| import OAI.Dynamics.ThreeTorus.Cutoffs |
| import OAI.Dynamics.ThreeTorus.NeutralSupport |
|
|
| namespace OAI |
|
|
| /-! Literal retained passage vectors, spectral identities, regularity, old marginals, and quantitative cutoff losses. -/ |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Set Filter |
| open scoped Topology |
| namespace ThreeTorus |
|
|
| lemma integral_passageChart {H : UnitAddCircle → UnitAddCircle} (hH : Measurable H) |
| (f : TripleTorus → ℝ) : |
| (∫ x, f (passageChart H x) ∂tripleVolume) = ∫ x, f x ∂tripleVolume := by |
| let e : TripleTorus ≃ᵐ TripleTorus := |
| { toEquiv := passageChart H |
| measurable_toFun := (measurePreserving_passageChart hH).measurable |
| measurable_invFun := passageChart_inverse_measurable hH } |
| exact (measurePreserving_passageChart hH).integral_comp e.measurableEmbedding f |
|
|
| / |
| The sole right-hand quantity is the actual summed Fourier-mass error. -/ |
| lemma physical_retainedPassage_error_le {m : ℤ} {H : UnitAddCircle → UnitAddCircle} |
| {A : UnitAddCircle × UnitAddCircle → ℂ} (hH : Continuous H) (hA : Continuous A) |
| (S : Finset ℤ) (χ W : ℤ → UnitAddCircle → ℝ) (hχc : ∀ j, Continuous (χ j)) |
| (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) |
| (hχ : ∀ j a, 0 ≤ χ j a ∧ χ j a ≤ 1) |
| (hW : ∀ j a, W j a ≠ 0 → j ∈ S ∧ χ j a = 1) |
| (hE : ∀ j, Integrable (fun a => |‖exactPassageCoeff m H A j a‖^2-W j a|) |
| AddCircle.haarAddCircle) |
| (hp : ∀ᵐ a ∂AddCircle.haarAddCircle, |
| Summable (fun j => |‖exactPassageCoeff m H A j a‖^2-W j a|)) |
| (hs : Summable (fun j => ∫ a, |‖exactPassageCoeff m H A j a‖^2-W j a| |
| ∂AddCircle.haarAddCircle)) : |
| (∫ x : TripleTorus, |
| ‖(ψ x.2.2:ℂ)*packet m A x-retainedPassage m H A S χ ψ (passageChart H x)‖^2 |
| ∂tripleVolume) ≤ |
| ∑' j : ℤ, ∫ a, |‖exactPassageCoeff m H A j a‖^2-W j a| ∂AddCircle.haarAddCircle := by |
| have he := integral_passageChart hH.measurable (fun x : TripleTorus => |
| ‖(ψ x.2.1:ℂ)*packet m A ((passageChart H).symm x)-retainedPassage m H A S χ ψ x‖^2) |
| simp only [Equiv.symm_apply_apply] at he |
| change (∫ x : TripleTorus, |
| ‖(ψ x.2.2:ℂ)*packet m A x-retainedPassage m H A S χ ψ (passageChart H x)‖^2 |
| ∂tripleVolume) = _ at he |
| rw [he,integrated_retainedPassage_error hH hA S χ hχc ψ hψc] |
| let E : UnitAddCircle → UnitAddCircle → ℝ := fun a θ => |
| ‖passageFourierInput m H A a θ- |
| ∑ j ∈ S, (χ j a:ℂ)*exactPassageCoeff m H A j a*fourier j θ‖^2 |
| have hEc : Continuous E.uncurry := by |
| apply Continuous.pow |
| apply Continuous.norm |
| apply (continuous_passageFourierInput hH hA).sub |
| apply continuous_finsetSum |
| intro j _hj |
| exact (((Complex.continuous_ofReal.comp (hχc j)).comp continuous_fst).mul |
| ((continuous_exactPassageCoeff hH hA j).comp continuous_fst)).mul |
| ((fourier j).continuous.comp continuous_snd) |
| have hIc : Continuous (fun a => ∫ θ, E a θ ∂AddCircle.haarAddCircle) := by |
| simpa only [setIntegral_univ] using |
| continuous_parametric_integral_of_continuous (μ := AddCircle.haarAddCircle) hEc isCompact_univ |
| have hIi : Integrable (fun a => ∫ θ, E a θ ∂AddCircle.haarAddCircle) AddCircle.haarAddCircle := |
| hIc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| calc |
| _ ≤ ∫ a, ∫ θ, E a θ ∂AddCircle.haarAddCircle ∂AddCircle.haarAddCircle := by |
| apply integral_mono_of_nonneg |
| · exact Filter.Eventually.of_forall (fun a => mul_nonneg (sq_nonneg _) |
| (integral_nonneg (fun _ => sq_nonneg _))) |
| · exact hIi |
| · filter_upwards [] with a |
| exact mul_le_of_le_one_left (integral_nonneg (fun _ => sq_nonneg _)) |
| (by nlinarith [(hψ a).1,(hψ a).2]) |
| _ ≤ _ := integrated_Fourier_cutoff_le_mass_error |
| (fun a => (continuous_passageFourierInput hH hA).comp (continuous_const.prodMk continuous_id)) S χ W hχ hW hE hp hs |
|
|
| lemma pointwise_array_mass_error_summable {c : ℤ → UnitAddCircle → ℂ} |
| {W : ℝ → UnitAddCircle → ℝ} (hc : HasCompactSupport W.uncurry) |
| (hcs : ∀ a, Summable (fun j => ‖c j a‖^2)) {n : ℕ} (hn : 0 < n) (a : UnitAddCircle) : |
| Summable (fun j : ℤ => |‖c j a‖^2-(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) := by |
| obtain ⟨S,hS⟩ := compact_lattice_support hc hn 0 |
| have hw : Summable (fun j : ℤ => |(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) := by |
| apply summable_of_ne_finset_zero (s := S) |
| intro j hj |
| have hz : W ((j:ℝ)/n) a = 0 := by simpa using hS j hj a |
| simp [hz] |
| refine Summable.of_nonneg_of_le (f := fun j => ‖c j a‖^2+|(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) |
| (fun _ => abs_nonneg _) (fun j => ?_) ((hcs a).add hw) |
| have hh := abs_sub_le (‖c j a‖^2) 0 ((n:ℝ)⁻¹*W ((j:ℝ)/n) a) |
| have he : |‖c j a‖^2| = ‖c j a‖^2 := abs_of_nonneg (sq_nonneg _) |
| simpa only [sub_zero,zero_sub,abs_neg,he] using hh |
|
|
| / |
| stationary mass theorem is applied. All summability hypotheses are derived. -/ |
| lemma physical_retainedPassage_tendsto |
| {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {W : ℝ → UnitAddCircle → ℝ} |
| (hH : ∀ n, Continuous (H n)) (hA : Continuous A) |
| (hWc : Continuous W.uncurry) (hWk : HasCompactSupport W.uncurry) |
| (S : ℕ → Finset ℤ) (χ : ℕ → ℤ → UnitAddCircle → ℝ) |
| (hχc : ∀ n j, Continuous (χ n j)) (hχ : ∀ n j a, 0 ≤ χ n j a ∧ χ n j a ≤ 1) |
| (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) |
| (hW : ∀ (n : ℕ), 0 < n → ∀ (j : ℤ) (a : UnitAddCircle), W ((j:ℝ)/n) a ≠ 0 → j ∈ S n ∧ χ n j a = 1) |
| (ht : Tendsto (fun n : ℕ => ∑' j : ℤ, ∫ a : UnitAddCircle, |
| |‖exactPassageCoeff m (H n) A j a‖^2-(n:ℝ)⁻¹*W ((j:ℝ)/n) a| |
| ∂AddCircle.haarAddCircle) atTop (𝓝 0)) : |
| Tendsto (fun n => ∫ x : TripleTorus, |
| ‖(ψ x.2.2:ℂ)*packet m A x- |
| retainedPassage m (H n) A (S n) (χ n) ψ (passageChart (H n) x)‖^2 |
| ∂tripleVolume) atTop (𝓝 0) := by |
| apply squeeze_zero' (Filter.Eventually.of_forall (fun _ => integral_nonneg (fun _ => sq_nonneg _))) _ ht |
| filter_upwards [eventually_gt_atTop 0] with n hn |
| have hEc (j : ℤ) : Continuous (fun a => |
| |‖exactPassageCoeff m (H n) A j a‖^2-(n:ℝ)⁻¹*W ((j:ℝ)/n) a|) := by |
| exact (((continuous_exactPassageCoeff (hH n) hA j).norm.pow 2).sub |
| (continuous_const.mul (hWc.comp (continuous_const.prodMk continuous_id)))).abs |
| apply physical_retainedPassage_error_le (hH n) hA (S n) (χ n) |
| (fun j a => (n:ℝ)⁻¹*W ((j:ℝ)/n) a) (hχc n) ψ hψc hψ (hχ n) |
| · intro j a ha |
| exact hW n hn j a (fun hz => ha (by simp [hz])) |
| · intro j |
| exact (hEc j).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| · apply Filter.Eventually.of_forall |
| intro a |
| exact pointwise_array_mass_error_summable hWk |
| (fun a => continuous_fourier_summable |
| ((continuous_passageFourierInput (hH n) hA).comp (continuous_const.prodMk continuous_id))) hn a |
| · simpa only [sub_zero] using integrated_array_mass_error_summable hWc hWk |
| (continuous_exactPassageCoeff (hH n) hA) (exactPassageCoeff_summable (hH n) hA) hn 0 |
|
|
| / |
| mass error is proved, not an input to this statement. -/ |
| lemma exact_physical_retainedPassage_tendsto |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} |
| {P V : ℝ → ℝ} {m : ℤ} {δ : ℕ → ℝ} {H : ℕ → UnitAddCircle → UnitAddCircle} |
| {o l u R : ℝ} |
| (hm : m ≠ 0) |
| (hH : ∀ n (x : ℝ), H n (x : UnitAddCircle) = ((n:ℝ)*(P x+δ n*x) : UnitAddCircle)) |
| (hA : ∀ a x, x ∈ Ioc o (o+1) → A ((x : UnitAddCircle),a) = A₀ a x) |
| (hA₀ : ∀ a, ContDiff ℝ 2 (A₀ a)) (hAc : Continuous A₀.uncurry) |
| (hAd : Continuous (fun p : UnitAddCircle × ℝ => deriv (A₀ p.1) p.2)) |
| (hsupp : ∀ a, Function.support (A₀ a) ⊆ Ioc l u) |
| (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) |
| (hP : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) P) |
| (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv P = V) |
| (hd : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) |
| (hHc : ∀ n, Continuous (H n)) (hAcont : Continuous A) (hδt : Tendsto δ atTop (𝓝 0)) |
| {χ : ℝ → UnitAddCircle → ℝ} (hχc : Continuous χ.uncurry) |
| (hχ : ∀ s a, 0 ≤ χ s a ∧ χ s a ≤ 1) |
| (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) |
| (hχW : ∀ s a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) |
| (A₀ a) s ≠ 0 → χ s a = 1) |
| (S : ℕ → Finset ℤ) |
| (hS : ∀ n, 0 < n → ∀ j ∉ S n, ∀ a, χ ((j:ℝ)/n) a = 0) : |
| Tendsto (fun n => ∫ x : TripleTorus, |
| ‖(ψ x.2.2:ℂ)*packet m A x- |
| retainedPassage m (H n) A (S n) (fun j => χ ((j:ℝ)/n)) ψ |
| (passageChart (H n) x)‖^2 ∂tripleVolume) atTop (𝓝 0) := by |
| have hbranch := scaled_speed_good_branch hm hR hV hd |
| have hW := stationaryWeight_signed_compact_continuous (contDiff_const.mul hV) hAc |
| (fun a => (hsupp a).trans Ioc_subset_Icc_self) hR hbranch.1 hbranch.2 |
| apply physical_retainedPassage_tendsto |
| (W := fun s a => stationaryWeight (Icc (l-R) (u+R)) |
| (fun x => (m:ℝ)*V x) (A₀ a) s) hHc hAcont hW.1 hW.2 S |
| (fun n j => χ ((j:ℝ)/n)) |
| (fun _ _ => hχc.comp (continuous_const.prodMk continuous_id)) |
| (fun _ _ _ => hχ _ _) ψ hψc hψ |
| · intro n hn j a hw |
| have he := hχW ((j:ℝ)/n) a hw |
| refine ⟨?_,he⟩ |
| by_contra hj |
| have hz := hS n hn j hj a |
| linarith |
| · exact exactPassageCoeff_summed hm hH hA hA₀ hAc hAd hsupp hlo huo hlu hR |
| hP hV hPV hd hHc hAcont hδt |
|
|
| end ThreeTorus |
|
|
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology |
| namespace ThreeTorus |
|
|
| lemma stationaryWeight_cylinder_support {C : Set ℝ} {V : ℝ → ℝ} {m : ℤ} |
| {A : UnitAddCircle → ℝ → ℂ} : |
| support (fun p : ℝ × UnitAddCircle => stationaryWeight C (fun x => (m:ℝ)*V x) (A p.2) p.1) ⊆ |
| stationaryImage m V (tsupport (fun p : ℝ × UnitAddCircle => A p.2 p.1)) := by |
| intro p hp |
| have hs : p.1 ∈ (fun x => (m:ℝ)*V x) '' C := by |
| by_contra hn |
| exact hp (by simp [stationaryWeight,branchDensity,hn]) |
| have hu := invFunOn_pos hs |
| have hA : A p.2 (invFunOn (fun x => (m:ℝ)*V x) C p.1) ≠ 0 := by |
| intro hz |
| exact hp (by simp [stationaryWeight,branchDensity,hs,hz]) |
| refine ⟨(invFunOn (fun x => (m:ℝ)*V x) C p.1,p.2),?_,Prod.ext hu.2 rfl⟩ |
| exact subset_tsupport _ hA |
|
|
| lemma lifted_amplitude_compactSupport {A : UnitAddCircle → ℝ → ℂ} {l u : ℝ} |
| (hA : ∀ a, support (A a) ⊆ Icc l u) : |
| HasCompactSupport (fun p : ℝ × UnitAddCircle => A p.2 p.1) ∧ |
| tsupport (fun p : ℝ × UnitAddCircle => A p.2 p.1) ⊆ Icc l u ×ˢ univ := by |
| have hs : tsupport (fun p : ℝ × UnitAddCircle => A p.2 p.1) ⊆ Icc l u ×ˢ univ := by |
| apply closure_minimal _ (isClosed_Icc.prod isClosed_univ) |
| intro p hp |
| exact ⟨hA p.2 hp,mem_univ _⟩ |
| exact ⟨(isCompact_Icc.prod isCompact_univ).of_isClosed_subset isClosed_closure hs,hs⟩ |
|
|
| / |
| shared angular index; asymptotic orthogonality is not used here. -/ |
| lemma retained_amplitude_disjoint {χ ξ : ℝ × UnitAddCircle → ℝ} |
| (hd : Disjoint (tsupport χ) (tsupport ξ)) (s : ℝ) |
| (F G : UnitAddCircle × UnitAddCircle → ℂ) : |
| Disjoint (tsupport (fun p : UnitAddCircle × UnitAddCircle => (χ (s,p.1):ℂ)*F p)) |
| (tsupport (fun p : UnitAddCircle × UnitAddCircle => (ξ (s,p.1):ℂ)*G p)) := by |
| have hs (η : ℝ × UnitAddCircle → ℝ) (B : UnitAddCircle × UnitAddCircle → ℂ) : |
| tsupport (fun p : UnitAddCircle × UnitAddCircle => (η (s,p.1):ℂ)*B p) ⊆ |
| (fun p : UnitAddCircle × UnitAddCircle => (s,p.1)) ⁻¹' tsupport η := by |
| apply closure_minimal _ ((isClosed_tsupport η).preimage (continuous_const.prodMk continuous_fst)) |
| intro p hp |
| exact subset_tsupport η (fun hz => hp (by simp [hz])) |
| exact (hd.preimage _).mono (hs χ F) (hs ξ G) |
|
|
| lemma norm_spare_character (m : ℤ) (C : UnitAddCircle → ℂ) (a b : UnitAddCircle) : |
| ‖C a * fourier m b‖ = ‖C a‖ := by |
| rw [norm_mul,show ‖fourier m b‖ = 1 from Circle.norm_coe _,mul_one] |
|
|
| / |
| zero-frequency exclusion and exact disjointness of the cutoffs. -/ |
| lemma colored_stationary_cutoffs {ι : Type*} [Finite ι] |
| (m : ι → ℤ) (V : ℝ → ℝ) (I : Set ℝ) |
| (K U : ι → Set (ℝ × UnitAddCircle)) |
| (hm : ∀ i, m i ≠ 0) (hV : Continuous V) (hi : InjOn V I) |
| (hK : ∀ i, IsCompact (K i)) (hKI : ∀ i, K i ⊆ I ×ˢ univ) |
| (hzero : ∀ x ∈ I, V x ≠ 0) |
| (hsep : ∀ i j, ∀ x ∈ I, ∀ y ∈ I, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) |
| (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (K i) (K j)) |
| (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, stationaryImage (m i) V (K i) ⊆ U i) : |
| ∃ χ : ι → ℝ × UnitAddCircle → ℝ, |
| (∀ i, CylinderSmooth (χ i) ∧ HasCompactSupport (χ i) ∧ tsupport (χ i) ⊆ U i ∧ |
| (∀ p, 0 ≤ χ i p ∧ χ i p ≤ 1) ∧ (∀ a, χ i (0,a) = 0) ∧ |
| ∀ p ∈ stationaryImage (m i) V (K i), χ i =ᶠ[𝓝 p] 1) ∧ |
| Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) := by |
| let E i := stationaryImage (m i) V (K i) |
| let O i := U i ∩ {p : ℝ × UnitAddCircle | p.1 ≠ 0} |
| have hO i : IsOpen (O i) := (hU i).inter (isOpen_ne_fun continuous_fst continuous_const) |
| have hEO i : E i ⊆ O i := by |
| intro p hp |
| refine ⟨hKU i hp,?_⟩ |
| obtain ⟨q,hq,rfl⟩ := hp |
| exact mul_ne_zero (by exact_mod_cast hm i) (hzero _ (hKI i hq).1) |
| have hEd : Pairwise (Disjoint on E) := by |
| intro i j hij |
| by_cases he : m i = m j |
| · change Disjoint (stationaryImage (m i) V (K i)) (stationaryImage (m j) V (K j)) |
| rw [he] |
| exact stationaryImage_disjoint_same (hm j) hi (hKI i) (hKI j) (hdisj i j hij he) |
| · apply stationaryImage_disjoint_distinct |
| intro p hp q hq hh |
| exact he (hsep i j p.1 (hKI i hp).1 q.1 (hKI j hq).1 hh) |
| obtain ⟨χ,hχ,hχd⟩ := finite_disjoint_cylinder_cutoffs E O |
| (fun i => stationaryImage_compact hV (hK i)) hO hEO hEd |
| refine ⟨χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_left, |
| (hχ i).2.2.2.1,?_,(hχ i).2.2.2.2⟩,hχd⟩ |
| intro a |
| by_contra ha |
| exact ((hχ i).2.2.1 (subset_tsupport (χ i) ha)).2 rfl |
|
|
| lemma simultaneous_retained_indices {ι : Type*} (χ : ι → ℝ × UnitAddCircle → ℝ) |
| (hc : ∀ i, HasCompactSupport (χ i)) (hz : ∀ i a, χ i (0,a) = 0) : |
| ∃ S : ℕ → ι → Finset ℤ, (∀ n i, (0:ℤ) ∉ S n i) ∧ |
| ∀ n, 0 < n → ∀ i j, j ∉ S n i → ∀ a, χ i ((j:ℝ)/n,a) = 0 := by |
| classical |
| have hex (n : ℕ) (i : ι) : ∃ S : Finset ℤ, (0:ℤ) ∉ S ∧ |
| (0 < n → ∀ j ∉ S, ∀ a, χ i ((j:ℝ)/n,a) = 0) := by |
| by_cases hn : 0 < n |
| · obtain ⟨S,hS,hS'⟩ := cylinder_cutoff_finite_indices (hc i) (hz i) n hn |
| exact ⟨S,hS,fun _ => hS'⟩ |
| · exact ⟨∅,Finset.notMem_empty _,fun h => (hn h).elim⟩ |
| choose S hS hS' using hex |
| exact ⟨S,hS,fun n hn i => hS' n i hn⟩ |
|
|
| / |
| The limit is a consequence of the signed stationary-phase theorem for the |
| actual Fourier coefficients, not an assumption on an auxiliary array. -/ |
| theorem colored_physical_passage {ι : Type*} [Finite ι] |
| {A : ι → UnitAddCircle × UnitAddCircle → ℂ} |
| {A₀ : ι → UnitAddCircle → ℝ → ℂ} |
| {P V : ℝ → ℝ} {m : ι → ℤ} {δ : ℕ → ℝ} |
| {H : ℕ → UnitAddCircle → UnitAddCircle} {o l u R : ℝ} |
| (hm : ∀ i, m i ≠ 0) |
| (hH : ∀ n (x : ℝ), H n (x : UnitAddCircle) = ((n:ℝ)*(P x+δ n*x) : UnitAddCircle)) |
| (hA : ∀ i a x, x ∈ Ioc o (o+1) → A i ((x : UnitAddCircle),a) = A₀ i a x) |
| (hA₀ : ∀ i a, ContDiff ℝ 2 (A₀ i a)) (hAc : ∀ i, Continuous (A₀ i).uncurry) |
| (hAd : ∀ i, Continuous (fun p : UnitAddCircle × ℝ => deriv (A₀ i p.1) p.2)) |
| (hsupp : ∀ i a, Function.support (A₀ i a) ⊆ Ioc l u) |
| (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) |
| (hP : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) P) |
| (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv P = V) |
| (hd : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) |
| (hHc : ∀ n, Continuous (H n)) (hAcont : ∀ i, Continuous (A i)) (hδt : Tendsto δ atTop (𝓝 0)) |
| (hzero : ∀ x ∈ Icc l u, V x ≠ 0) |
| (hsep : ∀ i j, ∀ x ∈ Icc l u, ∀ y ∈ Icc l u, |
| (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) |
| (hdisj : ∀ i j, i ≠ j → m i = m j → |
| Disjoint (tsupport (fun p : ℝ × UnitAddCircle => A₀ i p.2 p.1)) |
| (tsupport (fun p : ℝ × UnitAddCircle => A₀ j p.2 p.1))) |
| (U : ι → Set (ℝ × UnitAddCircle)) (hU : ∀ i, IsOpen (U i)) |
| (hKU : ∀ i, stationaryImage (m i) V |
| (tsupport (fun p : ℝ × UnitAddCircle => A₀ i p.2 p.1)) ⊆ U i) |
| (ψ : UnitAddCircle → ℝ) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) : |
| ∃ (χ : ι → ℝ × UnitAddCircle → ℝ) (S : ℕ → ι → Finset ℤ), |
| (∀ i, CylinderSmooth (χ i) ∧ HasCompactSupport (χ i) ∧ tsupport (χ i) ⊆ U i ∧ |
| (∀ p, 0 ≤ χ i p ∧ χ i p ≤ 1)) ∧ |
| Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ |
| (∀ n i, (0:ℤ) ∉ S n i) ∧ |
| (∀ n, 0 < n → ∀ i j, j ∉ S n i → ∀ a, χ i ((j:ℝ)/n,a) = 0) ∧ |
| ∀ i, Tendsto (fun n => ∫ x : TripleTorus, |
| ‖(ψ x.2.2:ℂ)*packet (m i) (A i) x- |
| retainedPassage (m i) (H n) (A i) (S n i) |
| (fun j a => χ i ((j:ℝ)/n,a)) ψ (passageChart (H n) x)‖^2 |
| ∂tripleVolume) atTop (𝓝 0) := by |
| let K i := tsupport (fun p : ℝ × UnitAddCircle => A₀ i p.2 p.1) |
| have hK i := lifted_amplitude_compactSupport |
| (fun a => (hsupp i a).trans Ioc_subset_Icc_self) |
| have hi : InjOn V (Icc l u) := by |
| apply (strictMonoOn_of_deriv_pos (convex_Icc _ _) hV.continuous.continuousOn _).injOn |
| intro x hx |
| have hx' := interior_subset hx |
| exact hd x ⟨by linarith [hx'.1],by linarith [hx'.2]⟩ |
| obtain ⟨χ,hχ,hχd⟩ := colored_stationary_cutoffs m V (Icc l u) K U hm hV.continuous hi |
| (fun i => (hK i).1) (fun i => (hK i).2) hzero hsep hdisj hU hKU |
| obtain ⟨S,hS,hS'⟩ := simultaneous_retained_indices χ (fun i => (hχ i).2.1) |
| (fun i => (hχ i).2.2.2.2.1) |
| refine ⟨χ,S,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1,(hχ i).2.2.2.1⟩, |
| hχd,hS,hS',?_⟩ |
| intro i |
| exact exact_physical_retainedPassage_tendsto (χ := fun s a => χ i (s,a)) (hm i) hH (hA i) (hA₀ i) (hAc i) |
| (hAd i) (hsupp i) hlo huo hlu hR hP hV hPV hd hHc (hAcont i) hδt |
| (hχ i).1.continuous (fun s a => (hχ i).2.2.2.1 (s,a)) ψ hψc hψ |
| (fun s a hw => ((hχ i).2.2.2.2.2 (s,a) |
| (stationaryWeight_cylinder_support hw)).self_of_nhds) |
| (fun n => S n i) (fun n hn j hj a => hS' n hn i j hj a) |
|
|
| end ThreeTorus |
|
|
| noncomputable section |
| open Set Function Filter |
| open scoped Topology |
| namespace ThreeTorus |
|
|
| / |
| alone, as opposed to the first passage's frequency/slow-coordinate cutoffs. -/ |
| theorem finite_disjoint_frequency_cutoffs {ι : Type*} [Finite ι] |
| (K U : ι → Set ℝ) (hK : ∀ i, IsCompact (K i)) (hU : ∀ i, IsOpen (U i)) |
| (hKU : ∀ i, K i ⊆ U i) (hdisj : Pairwise (Disjoint on K)) : |
| ∃ χ : ι → ℝ → ℝ, |
| (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ |
| tsupport (χ i) ⊆ U i ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ |
| ∀ s ∈ K i, χ i =ᶠ[𝓝 s] 1) ∧ |
| Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) := by |
| have hd : Pairwise (fun i j => Disjoint (𝓝ˢ (K i)) (𝓝ˢ (K j))) := |
| fun i j hij => disjoint_nhdsSet_nhdsSet (hK i).isClosed (hK j).isClosed (hdisj hij) |
| obtain ⟨O,hO,hOd⟩ := hd.exists_mem_filter_of_disjoint |
| have hv i : K i ⊆ interior (O i) ∩ U i := |
| subset_inter (subset_interior_iff_mem_nhdsSet.mpr (hO i)) (hKU i) |
| choose χ hχ using fun i => compact_smooth_cutoff (hK i) |
| (isOpen_interior.inter (hU i)) (hv i) |
| refine ⟨χ,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_right, |
| (hχ i).2.2.2⟩,?_⟩ |
| intro i j hij |
| exact (hOd hij).mono ((hχ i).2.2.1.trans (inter_subset_left.trans interior_subset)) |
| ((hχ j).2.2.1.trans (inter_subset_left.trans interior_subset)) |
|
|
| lemma compactSupport_frequency_cylinder {χ : ℝ → ℝ} (hc : HasCompactSupport χ) : |
| HasCompactSupport (fun p : ℝ × UnitAddCircle => χ p.1) := by |
| apply (hc.prod isCompact_univ).of_isClosed_subset isClosed_closure |
| apply closure_minimal _ ((isClosed_tsupport χ).prod isClosed_univ) |
| intro p hp |
| exact ⟨subset_tsupport χ hp,mem_univ _⟩ |
|
|
| / |
| sets. There is no asymptotic assertion in this exact separation conclusion. -/ |
| theorem disjoint_frequency_indices {ι : Type*} |
| (χ : ι → ℝ → ℝ) (hc : ∀ i, HasCompactSupport (χ i)) (hz : ∀ i, χ i 0 = 0) |
| (hd : Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j)))) : |
| ∃ S : ℕ → ι → Finset ℤ, |
| (∀ n i, (0:ℤ) ∉ S n i) ∧ |
| (∀ n, Pairwise (fun i j => Disjoint (S n i) (S n j))) ∧ |
| ∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0 := by |
| classical |
| have hex (n : ℕ) (i : ι) : ∃ S : Finset ℤ, |
| (∀ j ∈ S, χ i ((j:ℝ)/n) ≠ 0) ∧ |
| (0 < n → ∀ (j : ℤ), χ i ((j:ℝ)/n) ≠ 0 → j ∈ S) := by |
| by_cases hn : 0 < n |
| · obtain ⟨S,_hS,hS'⟩ := cylinder_cutoff_finite_indices |
| (compactSupport_frequency_cylinder (hc i)) (fun _ => hz i) n hn |
| refine ⟨S.filter (fun j => χ i ((j:ℝ)/n) ≠ 0),?_,?_⟩ |
| · intro j hj |
| exact (Finset.mem_filter.mp hj).2 |
| · intro _ j hj |
| apply Finset.mem_filter.mpr |
| exact ⟨by by_contra hh; exact hj (hS' j hh 0),hj⟩ |
| · exact ⟨∅,by simp,fun hh => (hn hh).elim⟩ |
| choose S hS hS' using hex |
| refine ⟨S,?_,?_,?_⟩ |
| · intro n i hi |
| exact hS n i 0 hi (by simp [hz]) |
| · intro n i j hij |
| rw [Finset.disjoint_left] |
| intro k hi hj |
| exact Set.disjoint_left.mp (hd hij) (subset_tsupport (χ i) (hS n i k hi)) |
| (subset_tsupport (χ j) (hS n j k hj)) |
| · intro n hn i j |
| exact ⟨hS n i j,hS' n i hn j⟩ |
|
|
| / |
| strictly monotone stationary maps. Distinct indices use the separated ranges. -/ |
| lemma frequency_images_disjoint {ι : Type*} (m : ι → ℤ) (V : ℝ → ℝ) |
| (I : Set ℝ) (K : ι → Set ℝ) (hm : ∀ i, m i ≠ 0) (hi : InjOn V I) |
| (hKI : ∀ i, K i ⊆ I) |
| (hsep : ∀ i j, ∀ x ∈ I, ∀ y ∈ I, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) |
| (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (K i) (K j)) : |
| Pairwise (fun i j => Disjoint ((fun x => (m i:ℝ)*V x) '' K i) |
| ((fun x => (m j:ℝ)*V x) '' K j)) := by |
| intro i j hij |
| rw [Set.disjoint_left] |
| intro z hz hz' |
| obtain ⟨x,hx,hxz⟩ := hz |
| obtain ⟨y,hy,hyz⟩ := hz' |
| have he : (m i:ℝ)*V x = (m j:ℝ)*V y := hxz.trans hyz.symm |
| have he' : m i = m j := hsep i j x (hKI i hx) y (hKI j hy) he |
| rw [he'] at he |
| have hm' : (m j:ℝ) ≠ 0 := by exact_mod_cast hm j |
| have hxy := hi (hKI i hx) (hKI j hy) (mul_left_cancel₀ hm' he) |
| exact Set.disjoint_left.mp (hdisj i j hij he') hx (hxy ▸ hy) |
|
|
| / |
| theorem second_separation_cutoffs {ι : Type*} [Finite ι] |
| (m : ι → ℤ) (V : ℝ → ℝ) (I : Set ℝ) (K U : ι → Set ℝ) |
| (hm : ∀ i, m i ≠ 0) (hV : Continuous V) (hi : InjOn V I) |
| (hK : ∀ i, IsCompact (K i)) (hKI : ∀ i, K i ⊆ I) |
| (hzero : ∀ x ∈ I, V x ≠ 0) |
| (hsep : ∀ i j, ∀ x ∈ I, ∀ y ∈ I, (m i:ℝ)*V x = (m j:ℝ)*V y → m i = m j) |
| (hdisj : ∀ i j, i ≠ j → m i = m j → Disjoint (K i) (K j)) |
| (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, (fun x => (m i:ℝ)*V x) '' K i ⊆ U i) : |
| ∃ (χ : ι → ℝ → ℝ) (S : ℕ → ι → Finset ℤ), |
| (∀ i, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (χ i) ∧ HasCompactSupport (χ i) ∧ |
| tsupport (χ i) ⊆ U i ∧ (∀ s, 0 ≤ χ i s ∧ χ i s ≤ 1) ∧ |
| ∀ s ∈ (fun x => (m i:ℝ)*V x) '' K i, χ i =ᶠ[𝓝 s] 1) ∧ |
| Pairwise (fun i j => Disjoint (tsupport (χ i)) (tsupport (χ j))) ∧ |
| (∀ n i, (0:ℤ) ∉ S n i) ∧ |
| (∀ n, Pairwise (fun i j => Disjoint (S n i) (S n j))) ∧ |
| ∀ n, 0 < n → ∀ i j, j ∈ S n i ↔ χ i ((j:ℝ)/n) ≠ 0 := by |
| let E i := (fun x => (m i:ℝ)*V x) '' K i |
| have hE i : IsCompact (E i) := (hK i).image (continuous_const.mul hV) |
| let O i := U i ∩ {x : ℝ | x ≠ 0} |
| have hO i : IsOpen (O i) := (hU i).inter isOpen_compl_singleton |
| have hEO i : E i ⊆ O i := by |
| intro z hz |
| refine ⟨hKU i hz,?_⟩ |
| obtain ⟨x,hx,rfl⟩ := hz |
| exact mul_ne_zero (by exact_mod_cast hm i) (hzero _ (hKI i hx)) |
| obtain ⟨χ,hχ,hχd⟩ := finite_disjoint_frequency_cutoffs E O hE hO hEO |
| (frequency_images_disjoint m V I K hm hi hKI hsep hdisj) |
| have hz i : χ i 0 = 0 := by |
| by_contra h |
| exact ((hχ i).2.2.1 (subset_tsupport (χ i) h)).2 rfl |
| obtain ⟨S,hS,hSd,hS'⟩ := disjoint_frequency_indices χ (fun i => (hχ i).2.1) hz hχd |
| exact ⟨χ,S,fun i => ⟨(hχ i).1,(hχ i).2.1,(hχ i).2.2.1.trans inter_subset_left, |
| (hχ i).2.2.2⟩,hχd,hS,hSd,hS'⟩ |
|
|
| end ThreeTorus |
|
|
| noncomputable section |
| open MeasureTheory Set Filter Function |
| open scoped Topology |
| namespace ThreeTorus |
| namespace CompactParameter |
|
|
| / |
| lemma hasFDerivAt_compact_integral |
| {E : Type} {G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] |
| {F : E × ℝ → G} (hF : ContDiff ℝ 1 F) (a b : ℝ) (x : E) : |
| HasFDerivAt (fun y => ∫ t in a..b, F (y,t)) |
| (∫ t in a..b, (fderiv ℝ F (x,t)).comp (ContinuousLinearMap.inl ℝ E ℝ)) x := by |
| let D : E × ℝ → E →L[ℝ] G := fun p => |
| (fderiv ℝ F p).comp (ContinuousLinearMap.inl ℝ E ℝ) |
| have hD : Continuous D := hF.continuous_fderiv (by norm_num) |>.clm_comp continuous_const |
| obtain ⟨C,hC⟩ := ((isCompact_closedBall x 1).prod (isCompact_uIcc (a := a) (b := b))).exists_bound_of_continuousOn hD.continuousOn |
| refine hasFDerivAt_integral_of_dominated_of_fderiv_le'' |
| (F := fun y t => F (y,t)) (F' := fun y t => D (y,t)) |
| (bound := fun _ => C) (Metric.closedBall_mem_nhds x (by norm_num : (0:ℝ)<1)) ?_ ?_ ?_ ?_ |
| intervalIntegrable_const ?_ |
| · exact Eventually.of_forall (fun y => (hF.continuous.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable) |
| · exact (hF.continuous.comp (continuous_const.prodMk continuous_id)).intervalIntegrable _ _ |
| · exact (hD.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| · filter_upwards [ae_restrict_mem measurableSet_uIoc] with t ht |
| intro y hy |
| exact hC (y,t) ⟨hy,uIoc_subset_uIcc ht⟩ |
| · exact Eventually.of_forall (fun t y _ => by |
| simpa [D,Function.comp_def] using ((hF.differentiable one_ne_zero ((ContinuousLinearMap.inl ℝ E ℝ) y+(0,t))).hasFDerivAt).comp y |
| ((ContinuousLinearMap.inl ℝ E ℝ).hasFDerivAt.add_const (0,t))) |
|
|
| / |
| lemma contDiff_compact_integral |
| {E : Type} {G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] |
| (n : ℕ) {F : E × ℝ → G} (hF : ContDiff ℝ n F) (a b : ℝ) : |
| ContDiff ℝ n (fun y => ∫ t in a..b, F (y,t)) := by |
| induction n generalizing G with |
| | zero => |
| apply contDiff_zero.mpr |
| exact intervalIntegral.continuous_parametric_intervalIntegral_of_continuous' |
| (show Continuous (Function.uncurry (fun y t => F (y,t))) from hF.continuous) a b |
| | succ n ih => |
| let D : E × ℝ → E →L[ℝ] G := fun p => |
| (fderiv ℝ F p).comp (ContinuousLinearMap.inl ℝ E ℝ) |
| have hD : ContDiff ℝ n D := (hF.fderiv_right (by simp)).clm_comp contDiff_const |
| have hder (x : E) := hasFDerivAt_compact_integral (hF.of_le (by exact_mod_cast Nat.succ_pos n)) a b x |
| rw [show ((n+1 : ℕ) : WithTop ℕ∞) = (n : WithTop ℕ∞)+1 by simp, contDiff_succ_iff_fderiv] |
| refine ⟨fun x => (hder x).differentiableAt, ?_, ?_⟩ |
| · simp |
| · have heq : fderiv ℝ (fun y => ∫ t in a..b, F (y,t)) = |
| fun y => ∫ t in a..b, D (y,t) := by |
| funext y |
| exact (hder y).fderiv |
| rw [heq] |
| exact ih hD |
|
|
| lemma smooth_compact_integral |
| {E : Type} {G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
| [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace G] |
| {F : E × ℝ → G} (hF : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) F) (a b : ℝ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun y => ∫ t in a..b, F (y,t)) := by |
| rw [contDiff_infty] |
| intro n |
| exact contDiff_compact_integral n (hF.of_le (by exact_mod_cast (le_top : (n : ℕ∞) ≤ ⊤))) a b |
|
|
| end CompactParameter |
|
|
| / |
| parameter, obtained from a compact interval integral of the literal lift. -/ |
| lemma smooth_fourierCoeff_param {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] {f : E → UnitAddCircle → ℂ} |
| (hf : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun p : E × ℝ => f p.1 (p.2 : UnitAddCircle))) (j : ℤ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => fourierCoeff (f x) j) := by |
| have hchar : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun p : E × ℝ => (fourier (-j) (p.2 : UnitAddCircle) : ℂ)) := by |
| simp only [fourier_coe_apply, Complex.ofReal_one, div_one] |
| exact (contDiff_const.mul (Complex.ofRealCLM.contDiff.comp contDiff_snd)).cexp |
| have h := CompactParameter.smooth_compact_integral (hchar.mul hf) 0 1 |
| have he (x : E) : fourierCoeff (f x) j = |
| ∫ t in (0:ℝ)..1, (fourier (-j) (t : UnitAddCircle) : ℂ) * f x t := by |
| simpa only [one_div_one,one_smul,zero_add,smul_eq_mul] using |
| fourierCoeff_eq_intervalIntegral (f x) j 0 |
| simpa only [he] using h |
|
|
| lemma smooth_fourier_lift (m : ℤ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun x : ℝ => (fourier m (x : UnitAddCircle) : ℂ)) := by |
| simp only [fourier_coe_apply,Complex.ofReal_one,div_one] |
| exact (contDiff_const.mul Complex.ofRealCLM.contDiff).cexp |
|
|
| / |
| approximation replaces these coefficients. -/ |
| lemma smooth_exactPassageCoeff {m : ℤ} {H : UnitAddCircle → UnitAddCircle} |
| {Hl : ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hHl : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Hl) |
| (hH : ∀ t : ℝ, H (t : UnitAddCircle) = (Hl t : UnitAddCircle)) |
| (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun p : ℝ × ℝ => A (p.1,p.2))) (j : ℤ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun a : ℝ => exactPassageCoeff m H A j a) := by |
| apply smooth_fourierCoeff_param |
| change ContDiff ℝ _ (fun p : ℝ × ℝ => A (p.2,p.1)*fourier m (H p.2)) |
| simp_rw [hH] |
| exact (hA.comp (contDiff_snd.prodMk contDiff_fst)).mul ((smooth_fourier_lift m).comp (hHl.comp contDiff_snd)) |
|
|
| / |
| class needed for its next use. -/ |
| lemma smooth_retained_amplitude {m j : ℤ} {H : UnitAddCircle → UnitAddCircle} |
| {Hl : ℝ → ℝ} {A : UnitAddCircle × UnitAddCircle → ℂ} |
| {χ : ℝ × UnitAddCircle → ℝ} {ψ : UnitAddCircle → ℝ} (s : ℝ) |
| (hHl : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Hl) |
| (hH : ∀ t : ℝ, H (t : UnitAddCircle) = (Hl t : UnitAddCircle)) |
| (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) |
| (hχ : CylinderSmooth χ) |
| (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun a : ℝ => ψ a)) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => |
| (χ (s,p.1):ℂ)*(ψ p.1:ℂ)*exactPassageCoeff m H A j p.1*fourier m (p.2:UnitAddCircle)) := by |
| have hc : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun p : ℝ × ℝ => χ (s,(p.1:UnitAddCircle))) := |
| hχ.comp (contDiff_const.prodMk contDiff_fst) |
| exact (((Complex.ofRealCLM.contDiff.comp hc).mul (Complex.ofRealCLM.contDiff.comp (hψ.comp contDiff_fst))).mul |
| ((smooth_exactPassageCoeff hHl hH hA j).comp contDiff_fst)).mul |
| ((smooth_fourier_lift m).comp contDiff_snd) |
|
|
| end ThreeTorus |
|
|
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology ENNReal |
| namespace ThreeTorus |
|
|
| / |
| measures is the identical ordinary measure identity. -/ |
| lemma positive_density_map_of_vectorMeasure |
| {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| {μ : Measure X} {ν : Measure Y} {f : X → Y} {q : X → ℝ} {p : Y → ℝ} |
| (hf : Measurable f) (hq : Integrable q μ) (hp : Integrable p ν) |
| (hq0 : ∀ x, 0 ≤ q x) (hp0 : ∀ y, 0 ≤ p y) |
| (he : (μ.withDensityᵥ q).map f = ν.withDensityᵥ p) : |
| (μ.withDensity (fun x => ENNReal.ofReal (q x))).map f = |
| ν.withDensity (fun y => ENNReal.ofReal (p y)) := by |
| ext s hs |
| rw [Measure.map_apply hf hs,withDensity_apply _ (hf hs),withDensity_apply _ hs] |
| rw [← ofReal_integral_eq_lintegral_ofReal hq.integrableOn (Eventually.of_forall hq0), |
| ← ofReal_integral_eq_lintegral_ofReal hp.integrableOn (Eventually.of_forall hp0)] |
| congr 1 |
| have h := congrArg (fun v : VectorMeasure Y ℝ => v s) he |
| simpa only [VectorMeasure.map_apply _ hf hs,withDensityᵥ_apply hq (hf hs), |
| withDensityᵥ_apply hp hs] using h |
|
|
| lemma branchDensity_nonneg {I : Set ℝ} {J g D q : ℝ → ℝ} |
| (hq : ∀ t, 0 ≤ q t) (y : ℝ) : 0 ≤ branchDensity I J g D q y := by |
| classical |
| simp only [branchDensity,Set.indicator_apply] |
| split_ifs |
| · exact div_nonneg (hq _) (abs_nonneg _) |
| · rfl |
|
|
| lemma unfoldedModeDensity_nonneg {I : Set ℝ} {J g D q : ℝ → ℝ} |
| (hq : ∀ t, 0 ≤ q t) {n : ℝ} (hn : 0 ≤ n) (j t : ℝ) : |
| 0 ≤ unfoldedModeDensity I J g D q n j t := by |
| unfold unfoldedModeDensity scaledBranchKernel |
| exact mul_nonneg (inv_nonneg.mpr (abs_nonneg _)) |
| (branchDensity_nonneg (fun a => mul_nonneg hn (hq a)) _) |
|
|
| lemma foldCircleDensity_nonneg {q : ℝ → ℝ} (hq : ∀ t, 0 ≤ q t) (z : UnitAddCircle) : |
| 0 ≤ foldCircleDensity q z := by |
| unfold foldCircleDensity periodizedReal |
| exact tsum_nonneg (fun k => hq _) |
|
|
| / |
| def retainedAmplitude (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) |
| (p : UnitAddCircle × UnitAddCircle) : ℂ := |
| (ψ p.1:ℂ)*(χ ((j:ℝ)/n) p.1:ℂ)*exactPassageCoeff m (H n) A j p.1*fourier m p.2 |
|
|
| lemma retainedPassage_eq_sum_packets (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (S : Finset ℤ) (n : ℕ) (x : TripleTorus) : |
| retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) ψ x = |
| ∑ j ∈ S, packet j (retainedAmplitude ψ χ m H A n j) x := rfl |
|
|
| lemma retainedAmplitude_norm_sq (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) |
| (p : UnitAddCircle × UnitAddCircle) : |
| ‖retainedAmplitude ψ χ m H A n j p‖^2 = |
| retainedCircleWeight ψ χ (fun n j => exactPassageCoeff m (H n) A j) n j p.1 := by |
| simp only [retainedAmplitude,norm_mul,show ‖fourier m p.2‖ = 1 from Circle.norm_coe _, |
| mul_one,mul_pow,Complex.norm_real,Real.norm_eq_abs,sq_abs,retainedCircleWeight] |
|
|
| lemma retainedAmplitude_modulus (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (a b : UnitAddCircle) : |
| ‖retainedAmplitude ψ χ m H A n j (a,b)‖ = |
| ‖retainedAmplitude ψ χ m H A n j (a,0)‖ := by |
| simp only [retainedAmplitude,norm_mul,show ‖fourier m b‖ = 1 from Circle.norm_coe _, |
| show ‖fourier m (0:UnitAddCircle)‖ = 1 from Circle.norm_coe _] |
|
|
| |
| lemma map_withDensity_comp {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) {ζ : X → Y} {w : Y → ℝ≥0∞} |
| (hζ : Measurable ζ) (hw : Measurable w) : |
| (μ.withDensity (w ∘ ζ)).map ζ = (μ.map ζ).withDensity w := by |
| ext s hs |
| rw [Measure.map_apply hζ hs, withDensity_apply _ (hζ hs), withDensity_apply _ hs] |
| rw [← lintegral_indicator (hζ hs), ← lintegral_indicator hs, |
| lintegral_map (hw.indicator hs) hζ] |
| apply lintegral_congr |
| intro x |
| rfl |
|
|
| / |
| literal packet spectral measure is the single-coordinate weighted image. -/ |
| lemma packetSpectralMeasure_of_modulus |
| {φ : UnitAddCircle → UnitAddCircle} (hφ : Measurable φ) {j : ℤ} |
| {B : UnitAddCircle × UnitAddCircle → ℂ} {q : UnitAddCircle → ℝ} |
| (hq : Measurable q) (hB : ∀ p, ‖B p‖^2 = q p.1) : |
| packetSpectralMeasure φ j B = |
| (AddCircle.haarAddCircle.withDensity (fun a => ENNReal.ofReal (q a))).map |
| (fun a => j • φ a) := by |
| unfold packetSpectralMeasure |
| have he : (fun p : UnitAddCircle × UnitAddCircle => ENNReal.ofReal (‖B p‖^2)) = |
| (fun a => ENNReal.ofReal (q a)) ∘ Prod.fst := funext (fun p => congrArg _ (hB p)) |
| rw [he] |
| change Measure.map ((fun a => j • φ a) ∘ Prod.fst) _ = _ |
| rw [← Measure.map_map (show Measurable (fun a => j • φ a) from hφ.const_smul j) measurable_fst, |
| map_withDensity_comp slowVolume measurable_fst hq.ennreal_ofReal] |
| simp only [slowVolume,Measure.map_fst_prod,measure_univ,one_smul] |
|
|
| lemma retainedCircleWeight_nonneg (ψ : UnitAddCircle → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (c : ℕ → ℤ → UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (a : UnitAddCircle) : |
| 0 ≤ retainedCircleWeight ψ χ c n j a := by unfold retainedCircleWeight; positivity |
|
|
| lemma circleDensityLift_nonneg {q : UnitAddCircle → ℝ} (hq : ∀ a, 0 ≤ q a) |
| (o x : ℝ) : 0 ≤ circleDensityLift o q x := by |
| classical |
| simp only [circleDensityLift,Set.indicator_apply] |
| split_ifs |
| · exact hq _ |
| · rfl |
|
|
| / |
| an identity of signed integrals. -/ |
| lemma positive_foldedModeDensity_pushforward {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} |
| {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} |
| (hq : Integrable q AddCircle.haarAddCircle) (hqm : Measurable q₀) |
| (hq0 : ∀ a, 0 ≤ q a) (hq00 : ∀ x, 0 ≤ q₀ x) |
| (hq₀ : ∀ x ∈ Ioc o (o+1), q (x : UnitAddCircle) = q₀ x) |
| (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) |
| (hj : Measurable J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) |
| (hJe : ∀ x ∈ I, J (x : UnitAddCircle) = J₀ x) |
| (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) |
| {n j : ℝ} (hn : 0 < n) (hjn : j ≠ 0) : |
| (AddCircle.haarAddCircle.withDensity (fun a => ENNReal.ofReal (q a))).map |
| (fun a => (((j+(j/n)*J a)/n : ℝ) : UnitAddCircle)) = |
| AddCircle.haarAddCircle.withDensity |
| (fun z => ENNReal.ofReal |
| (foldCircleDensity (unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ n j) z)) := by |
| have hqi : Integrable q₀ := by |
| rw [← circleDensityLift_eq hq₀ hq₀s] |
| exact integrable_circleDensityLift hq o |
| have hD : ∀ x ∈ I, HasDerivWithinAt J₀ (deriv J₀ x) I x := |
| fun x _ => (hJ.differentiable (by simp) x).hasDerivAt.hasDerivWithinAt |
| have hmi := measurable_unfoldedModeDensity hJ hIo hinj hd hqm n j |
| have hii := integrable_unfoldedModeDensity hIo.measurableSet |
| (isOpen_image_of_deriv_ne_zero hJ hIo hd).measurableSet hD hinj |
| (fun _ hx => hinj.leftInvOn_invFunOn hx) hd hqi hn hjn |
| exact positive_density_map_of_vectorMeasure |
| (AddCircle.measurable_mk'.comp ((measurable_const.add (hj.const_mul _)).div_const n)) |
| hq (integrable_foldCircleDensity hmi hii) hq0 |
| (foldCircleDensity_nonneg (unfoldedModeDensity_nonneg hq00 hn.le j)) |
| (foldedModeDensity_pushforward hq hqm hq₀ hq₀s hqs hj hJ hJe hIo hinj hd hn hjn) |
|
|
| lemma passage_speed_mode (J : UnitAddCircle → ℝ) (n : ℝ) (j : ℤ) (a : UnitAddCircle) : |
| j • (((1+J a/n)/n : ℝ) : UnitAddCircle) = |
| ((((j:ℝ)+((j:ℝ)/n)*J a)/n : ℝ) : UnitAddCircle) := by |
| rw [← AddCircle.coe_zsmul] |
| congr 1 |
| simp only [zsmul_eq_mul] |
| ring |
|
|
| / |
| nonnegative folded density under the actual new twist. -/ |
| lemma packetSpectralMeasure_new_speed {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} |
| {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} |
| {B : UnitAddCircle × UnitAddCircle → ℂ} |
| (hq : Integrable q AddCircle.haarAddCircle) (hqmeas : Measurable q) (hqm : Measurable q₀) |
| (hB : ∀ p, ‖B p‖^2 = q p.1) (hq00 : ∀ x, 0 ≤ q₀ x) |
| (hq₀ : ∀ x ∈ Ioc o (o+1), q (x : UnitAddCircle) = q₀ x) |
| (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) |
| (hj : Measurable J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) |
| (hJe : ∀ x ∈ I, J (x : UnitAddCircle) = J₀ x) |
| (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) |
| {n : ℝ} {j : ℤ} (hn : 0 < n) (hjn : j ≠ 0) : |
| packetSpectralMeasure (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) j B = |
| AddCircle.haarAddCircle.withDensity |
| (fun z => ENNReal.ofReal |
| (foldCircleDensity (unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ n j) z)) := by |
| have hσ : Measurable (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) := |
| AddCircle.measurable_mk'.comp ((measurable_const.add (hj.div_const n)).div_const n) |
| rw [packetSpectralMeasure_of_modulus hσ hqmeas hB] |
| simp_rw [passage_speed_mode] |
| exact positive_foldedModeDensity_pushforward hq hqm |
| (fun a => (hB (a,0)) ▸ sq_nonneg _) hq00 hq₀ hq₀s hqs hj hJ hJe hIo hinj hd hn |
| (by exact_mod_cast hjn) |
|
|
| / |
| The only support hypothesis is that the cutoff lies on the chosen good branch. -/ |
| lemma retainedAmplitude_spectralDensity |
| {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} |
| {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} |
| {J : UnitAddCircle → ℝ} {J₀ : ℝ → ℝ} {o : ℝ} {I : Set ℝ} |
| (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) |
| (hH : ∀ n, Continuous (H n)) (hA : Continuous A) |
| (hψs : ∀ x ∈ Ioc o (o+1), ψ (x:UnitAddCircle) ≠ 0 → x ∈ I) |
| (hj : Measurable J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) |
| (hJe : ∀ x ∈ I, J (x:UnitAddCircle) = J₀ x) |
| (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) |
| {n : ℕ} {j : ℤ} (hn : 0 < n) (hjn : j ≠ 0) : |
| packetSpectralMeasure (fun a => (((1+J a/n)/n:ℝ):UnitAddCircle)) j |
| (retainedAmplitude ψ χ m H A n j) = |
| AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal |
| (foldCircleDensity (unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) |
| (circleDensityLift o (retainedCircleWeight ψ χ |
| (fun n j => exactPassageCoeff m (H n) A j) n j)) n j) z)) := by |
| let C n j := exactPassageCoeff m (H n) A j |
| let q := retainedCircleWeight ψ χ C n j |
| have hC n j : Continuous (C n j) := continuous_exactPassageCoeff (hH n) hA j |
| have hq : Continuous q := |
| ((hψ.pow 2).mul ((hχ.comp (continuous_const.prodMk continuous_id)).pow 2)).mul |
| ((hC n j).norm.pow 2) |
| have hqs : support (circleDensityLift o q) ⊆ Ioc o (o+1) ∩ I := by |
| intro x hx |
| have hxI : x ∈ Ioc o (o+1) := by |
| by_contra hh |
| exact hx (by simp [circleDensityLift,hh]) |
| refine ⟨hxI,hψs x hxI ?_⟩ |
| intro hz |
| exact hx (by simp [circleDensityLift,hxI,q,retainedCircleWeight,hz]) |
| exact packetSpectralMeasure_new_speed |
| (retainedCircleWeight_integrable hψ hχ hC n j) hq.measurable |
| (measurable_circleDensityLift hq.measurable o) |
| (retainedAmplitude_norm_sq ψ χ m H A n j) |
| (circleDensityLift_nonneg (retainedCircleWeight_nonneg ψ χ C n j) o) |
| (fun _ hx => by simp only [circleDensityLift,indicator_of_mem hx]; rfl) |
| (hqs.trans inter_subset_left) (hqs.trans inter_subset_right) |
| hj hJ hJe hIo hinj hd (by exact_mod_cast hn) hjn |
|
|
| lemma packet_twisted_pair_zero {A B : UnitAddCircle × UnitAddCircle → ℂ} |
| (hA : Continuous A) (hB : Continuous B) |
| {φ : UnitAddCircle → UnitAddCircle} (hφ : Continuous φ) {m k : ℤ} |
| (hdisj : m = k → Disjoint (support A) (support B)) (n : ℤ) : |
| (∫ x : TripleTorus, starRingEnd ℂ (packet m A x)*packet k B (twistIter φ n x) |
| ∂tripleVolume) = 0 := by |
| let B' : UnitAddCircle × UnitAddCircle → ℂ := fun p => fourier n (k • φ p.1)*B p |
| have hb' : Continuous B' := |
| ((fourier n).continuous.comp ((hφ.comp continuous_fst).const_smul k)).mul hB |
| have he (x : TripleTorus) : packet k B (twistIter φ n x) = packet k B' x := by |
| rw [packet_twistIter] |
| dsimp [packet,B'] |
| ring |
| simp_rw [he] |
| rw [integral_packet_pair hA hb'] |
| split_ifs with hmk |
| · apply integral_eq_zero_of_ae |
| filter_upwards [] with p |
| by_cases ha : A p = 0 |
| · simp [ha] |
| · have hb : B p = 0 := by |
| by_contra hh |
| exact Set.disjoint_left.mp (hdisj hmk) ha hh |
| simp [B',hb] |
| · rfl |
|
|
| / |
| literal packet pushforwards, demonstrated by every bilateral Fourier moment. -/ |
| lemma disjoint_packet_list_correlation {ι : Type*} [Fintype ι] |
| (m : ι → ℤ) (A : ι → UnitAddCircle × UnitAddCircle → ℂ) |
| (hA : ∀ i, Continuous (A i)) {φ : UnitAddCircle → UnitAddCircle} (hφ : Continuous φ) |
| (hd : ∀ i j, i ≠ j → m i = m j → Disjoint (support (A i)) (support (A j))) |
| (n : ℤ) : |
| (∫ x : TripleTorus, starRingEnd ℂ (∑ i, packet (m i) (A i) x)* |
| (∑ i, packet (m i) (A i) (twistIter φ n x)) ∂tripleVolume) = |
| ∑ i, ∫ z, fourier n z ∂packetSpectralMeasure φ (m i) (A i) := by |
| classical |
| have ht : Continuous (twistIter φ n) := |
| (continuous_fst.add ((hφ.comp (continuous_fst.comp continuous_snd)).const_smul n)).prodMk |
| continuous_snd |
| have hc (i j : ι) : Continuous (fun x : TripleTorus => |
| starRingEnd ℂ (packet (m i) (A i) x)*packet (m j) (A j) (twistIter φ n x)) := |
| (Complex.continuous_conj.comp (continuous_packet (hA i))).mul |
| ((continuous_packet (hA j)).comp ht) |
| have hi (i j : ι) : Integrable (fun x : TripleTorus => |
| starRingEnd ℂ (packet (m i) (A i) x)*packet (m j) (A j) (twistIter φ n x)) tripleVolume := |
| (hc i j).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| simp_rw [map_sum,Finset.sum_mul,Finset.mul_sum] |
| rw [integral_finsetSum _ (fun i _ => integrable_finsetSum _ (fun j _ => hi i j))] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [integral_finsetSum _ (fun j _ => hi i j),Finset.sum_eq_single i] |
| · exact (packet_spectral_moment hφ (hA i) (m i) n).symm |
| · intro j _ hji |
| exact packet_twisted_pair_zero (hA i) (hA j) hφ (hd i j hji.symm) n |
| · intro hi |
| exact (hi (Finset.mem_univ i)).elim |
|
|
| end ThreeTorus |
| end |
| end |
| end |
| end |
| end |
| end |
|
|
| section |
|
|
| /-! Exact spectral identity for the literal retained packet vector. This closes |
| the interface between the previously proved coefficient-density estimates and |
| the actual Koopman correlations, without an abstract density hypothesis. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators |
| namespace ThreeTorus |
|
|
| lemma foldCircleDensity_add_ae {q r : ℝ → ℝ} (hq : Measurable q) (hr : Measurable r) |
| (hqi : Integrable q) (hri : Integrable r) : |
| foldCircleDensity (fun x => q x+r x) =ᵐ[AddCircle.haarAddCircle] |
| fun z => foldCircleDensity q z+foldCircleDensity r z := by |
| apply (integrable_foldCircleDensity (hq.add hr) (hqi.add hri)).ae_eq_of_withDensityᵥ_eq |
| ((integrable_foldCircleDensity hq hqi).add (integrable_foldCircleDensity hr hri)) |
| rw [← foldCircleDensity_pushforward (hq.add hr) (hqi.add hri), |
| withDensityᵥ_add hqi hri, |
| withDensityᵥ_add (integrable_foldCircleDensity hq hqi) (integrable_foldCircleDensity hr hri)] |
| have he := map_add (VectorMeasure.mapGm (fun x : ℝ => (x:UnitAddCircle))) |
| (volume.withDensityᵥ q) (volume.withDensityᵥ r) |
| change (volume.withDensityᵥ q+volume.withDensityᵥ r).map _ = |
| (volume.withDensityᵥ q).map _+(volume.withDensityᵥ r).map _ at he |
| rw [he,foldCircleDensity_pushforward hq hqi,foldCircleDensity_pushforward hr hri] |
|
|
| lemma foldCircleDensity_finsetSum_ae {ι : Type*} (S : Finset ι) (q : ι → ℝ → ℝ) |
| (hq : ∀ i ∈ S, Measurable (q i)) (hqi : ∀ i ∈ S, Integrable (q i)) : |
| foldCircleDensity (fun x => ∑ i ∈ S, q i x) =ᵐ[AddCircle.haarAddCircle] |
| fun z => ∑ i ∈ S, foldCircleDensity (q i) z := by |
| classical |
| induction S using Finset.induction_on with |
| | empty => |
| filter_upwards [] with z |
| simp [foldCircleDensity,periodizedReal,AddCircle.liftIoc,Set.domRestrict] |
| | @insert i S hi ih => |
| simp only [Finset.sum_insert hi] |
| have hh := foldCircleDensity_add_ae (hq i (Finset.mem_insert_self i S)) |
| (Finset.measurable_sum S (fun j hj => hq j (Finset.mem_insert_of_mem hj))) |
| (hqi i (Finset.mem_insert_self i S)) |
| (integrable_finsetSum S (fun j hj => hqi j (Finset.mem_insert_of_mem hj))) |
| have ht := ih (fun j hj => hq j (Finset.mem_insert_of_mem hj)) |
| (fun j hj => hqi j (Finset.mem_insert_of_mem hj)) |
| filter_upwards [hh,ht] with z hz ht |
| rw [hz,ht] |
|
|
| lemma moment_withDensity_real {p : UnitAddCircle → ℝ} (hp : Measurable p) |
| (hp0 : ∀ z, 0 ≤ p z) (v : UnitAddCircle → ℂ) : |
| (∫ z, v z ∂AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (p z))) = |
| ∫ z, (p z:ℂ)*v z ∂AddCircle.haarAddCircle := by |
| rw [integral_withDensity_eq_integral_toReal_smul hp.ennreal_ofReal |
| (by filter_upwards [] with z; finiteness)] |
| apply integral_congr_ae |
| filter_upwards [] with z |
| rw [ENNReal.toReal_ofReal (hp0 z),Complex.real_smul] |
|
|
| lemma continuous_retainedAmplitude |
| {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} |
| {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) |
| (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (n : ℕ) (j : ℤ) : |
| Continuous (retainedAmplitude ψ χ m H A n j) := by |
| unfold retainedAmplitude |
| exact ((((Complex.continuous_ofReal.comp hψ).comp continuous_fst).mul |
| (Complex.continuous_ofReal.comp (hχ.comp (continuous_const.prodMk continuous_fst)))).mul |
| ((continuous_exactPassageCoeff (hH n) hA j).comp continuous_fst)).mul |
| ((fourier m).continuous.comp continuous_snd) |
|
|
| / |
| exactly those of its explicit nonnegative folded coefficient density. -/ |
| theorem retainedPassage_spectral_moment |
| {ψ : UnitAddCircle → ℝ} {χ : ℝ → UnitAddCircle → ℝ} |
| {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} |
| {J : UnitAddCircle → ℝ} {J₀ : ℝ → ℝ} {I : Set ℝ} |
| (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) |
| (hH : ∀ n, Continuous (H n)) (hA : Continuous A) |
| (hψs : ∀ x ∈ Ioc (0:ℝ) 1, ψ (x:UnitAddCircle) ≠ 0 → x ∈ I) |
| (hj : Continuous J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) |
| (hJe : ∀ x ∈ I, J (x:UnitAddCircle) = J₀ x) |
| (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) |
| {n : ℕ} {S : Finset ℤ} (hn : 0 < n) (hS : ∀ j ∈ S, j ≠ 0) (k : ℤ) : |
| let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) |
| let f := retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) ψ |
| (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter φ k x) ∂tripleVolume) = |
| ∫ z, (exactRetainedPacketDensity I J₀ ψ χ m H A S n z:ℂ)*fourier k z |
| ∂AddCircle.haarAddCircle := by |
| classical |
| dsimp only |
| let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) |
| let B := retainedAmplitude ψ χ m H A n |
| let q := fun j : ℤ => unfoldedModeDensity I J₀ (invFunOn J₀ I) (deriv J₀) |
| (circleDensityLift 0 (retainedCircleWeight ψ χ |
| (fun n j => exactPassageCoeff m (H n) A j) n j)) n j |
| have hφ : Continuous φ := |
| (AddCircle.continuous_mk' 1).comp ((continuous_const.add (hj.div_const n)).div_const n) |
| have hB j : Continuous (B j) := continuous_retainedAmplitude hψ hχ hH hA n j |
| have hreg j (hjS : j ∈ S) := retained_unfolded_regular hJ hIo hinj hd hψ hχ |
| (fun n j => continuous_exactPassageCoeff (m := m) (hH n) hA j) 0 hn (hS j hjS) |
| have hqm j (hjS : j ∈ S) : Measurable (q j) := (hreg j hjS).1 |
| have hqi j (hjS : j ∈ S) : Integrable (q j) := (hreg j hjS).2 |
| have hq0 j : ∀ t, 0 ≤ q j t := |
| unfoldedModeDensity_nonneg (circleDensityLift_nonneg |
| (retainedCircleWeight_nonneg ψ χ _ n j) 0) (Nat.cast_nonneg n) j |
| have hds j (hjS : j ∈ S) : packetSpectralMeasure φ j (B j) = |
| AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal (foldCircleDensity (q j) z)) := |
| retainedAmplitude_spectralDensity hψ hχ hH hA (by simpa using hψs) hj.measurable |
| hJ hJe hIo hinj hd hn (hS j hjS) |
| have hfold := foldCircleDensity_finsetSum_ae S q hqm hqi |
| have hint j (hjS : j ∈ S) : Integrable |
| (fun z => (foldCircleDensity (q j) z:ℂ)*fourier k z) AddCircle.haarAddCircle := |
| (integrable_foldCircleDensity (hqm j hjS) (hqi j hjS)).ofReal.mul_bdd |
| (fourier k).continuous.aestronglyMeasurable |
| (ae_of_all _ (fun z => le_of_eq (Circle.norm_coe _))) |
| have hc := disjoint_packet_list_correlation (fun j : S => (j:ℤ)) |
| (fun j : S => B j) (fun j => hB j) hφ |
| (fun i j hij he => (hij (Subtype.ext he)).elim) k |
| have hsum (x : TripleTorus) : (∑ j : S, packet (j:ℤ) (B j) x) = |
| ∑ j ∈ S, packet j (B j) x := Finset.sum_coe_sort S (fun j => packet j (B j) x) |
| simp_rw [hsum] at hc |
| rw [Finset.sum_coe_sort S (fun j => ∫ z, fourier k z ∂packetSpectralMeasure φ j (B j))] at hc |
| simp only [retainedPassage_eq_sum_packets] |
| change (∫ x : TripleTorus, starRingEnd ℂ (∑ j ∈ S, packet j (B j) x)* |
| (∑ j ∈ S, packet j (B j) (twistIter φ k x)) ∂tripleVolume) = _ |
| rw [hc] |
| calc |
| _ = ∑ j ∈ S, ∫ z, (foldCircleDensity (q j) z:ℂ)*fourier k z |
| ∂AddCircle.haarAddCircle := by |
| apply Finset.sum_congr rfl |
| intro j hjS |
| rw [hds j hjS] |
| exact moment_withDensity_real (measurable_foldCircleDensity (hqm j hjS)) |
| (foldCircleDensity_nonneg (hq0 j)) (fourier k) |
| _ = ∫ z, (∑ j ∈ S, (foldCircleDensity (q j) z:ℂ))*fourier k z |
| ∂AddCircle.haarAddCircle := by |
| simp_rw [Finset.sum_mul] |
| exact (integral_finsetSum _ hint).symm |
| _ = _ := by |
| apply integral_congr_ae |
| filter_upwards [hfold] with z hz |
| change (∑ j ∈ S, (foldCircleDensity (q j) z:ℂ))*fourier k z = |
| (foldCircleDensity (fun t => ∑ j ∈ S, q j t) z:ℂ)*fourier k z |
| rw [hz,Complex.ofReal_sum] |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! Smoothness at the edge of the good arc is essential: the actual retained |
| spectral densities must be valid smooth inputs to the next stage. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators |
| namespace ThreeTorus |
|
|
| lemma support_branchDensity_subset {J q : ℝ → ℝ} {I : Set ℝ} : |
| support (branchDensity I J (invFunOn J I) (deriv J) q) ⊆ J '' tsupport q := by |
| intro y hy |
| have hyI : y ∈ J '' I := by |
| by_contra hh |
| exact hy (by simp [branchDensity,hh]) |
| have hx := invFunOn_pos hyI |
| refine ⟨invFunOn J I y,subset_tsupport q ?_,hx.2⟩ |
| intro hq |
| exact hy (by simp [branchDensity,hyI,hq]) |
|
|
| lemma branchDensity_compact {J q : ℝ → ℝ} {I : Set ℝ} |
| (hJ : Continuous J) (hq : HasCompactSupport q) : |
| HasCompactSupport (branchDensity I J (invFunOn J I) (deriv J) q) := by |
| apply HasCompactSupport.intro (hq.image hJ) |
| intro y hy |
| exact notMem_support.mp (fun hh => hy (support_branchDensity_subset hh)) |
|
|
| lemma branchDensity_smooth {J q : ℝ → ℝ} {I : Set ℝ} |
| (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) |
| (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) |
| (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) q) (hc : HasCompactSupport q) |
| (hs : tsupport q ⊆ I) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (branchDensity I J (invFunOn J I) (deriv J) q) := by |
| classical |
| let g := invFunOn J I |
| have hM : IsCompact (J '' tsupport q) := hc.image hJ.continuous |
| have hD : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (deriv J) := hJ.deriv' |
| rw [contDiff_iff_contDiffAt] |
| intro y |
| by_cases hy : y ∈ J '' tsupport q |
| · obtain ⟨x,hx,rfl⟩ := hy |
| have hxI := hs hx |
| have hyI : J x ∈ J '' I := mem_image_of_mem J hxI |
| have hg := contDiffAt_invFunOn_of_deriv_ne_zero hJ hIo hinj hxI |
| ((hJ.differentiable (by simp) x).hasDerivAt) (hd x hxI) |
| have hge : g (J x) = x := hinj (invFunOn_pos hyI).1 hxI (invFunOn_pos hyI).2 |
| have hden : ContDiffAt ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun y => |deriv J (g y)|) (J x) := |
| (contDiffAt_abs (by simpa only [hge] using hd x hxI)).comp (J x) |
| (hD.contDiffAt.comp (J x) hg) |
| have hquot := (hq.contDiffAt.comp (J x) hg).div hden |
| (by simpa only [hge] using abs_ne_zero.mpr (hd x hxI)) |
| apply hquot.congr_of_eventuallyEq |
| filter_upwards [(isOpen_image_of_deriv_ne_zero hJ hIo hd).mem_nhds hyI] with z hz |
| simp only [branchDensity,indicator_of_mem hz,g] |
| rfl |
| · apply (contDiffAt_const (c := (0:ℝ))).congr_of_eventuallyEq |
| filter_upwards [hM.isClosed.isOpen_compl.mem_nhds hy] with z hz |
| exact notMem_support.mp (fun hh => hz (support_branchDensity_subset hh)) |
|
|
| lemma unfoldedModeDensity_smooth {J q : ℝ → ℝ} {I : Set ℝ} |
| (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) |
| (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) |
| (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) q) (hc : HasCompactSupport q) |
| (hs : tsupport q ⊆ I) (n j : ℝ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (unfoldedModeDensity I J (invFunOn J I) (deriv J) q n j) := by |
| have hmul : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x => n*q x) := contDiff_const.mul hq |
| have hcs : HasCompactSupport (fun x => n*q x) := hc.mul_left |
| have hs' : tsupport (fun x => n*q x) ⊆ I := tsupport_mul_subset_right.trans hs |
| have hh := branchDensity_smooth hJ hIo hinj hd hmul hcs hs' |
| unfold unfoldedModeDensity scaledBranchKernel |
| exact contDiff_const.mul (hh.comp ((contDiff_const.mul contDiff_id).sub contDiff_const |>.div_const _)) |
|
|
| lemma unfoldedModeDensity_compact {J q : ℝ → ℝ} {I : Set ℝ} |
| (hJ : Continuous J) (hc : HasCompactSupport q) {n j : ℝ} (hn : n ≠ 0) (hj : j ≠ 0) : |
| HasCompactSupport (unfoldedModeDensity I J (invFunOn J I) (deriv J) q n j) := by |
| have hh := branchDensity_compact (I := I) hJ (hc.mul_left (f := fun _ => n)) |
| let e : ℝ ≃ₜ ℝ := (Homeomorph.mulLeft₀ n hn).trans (Homeomorph.addRight (-j)) |
| let e' : ℝ ≃ₜ ℝ := e.trans (Homeomorph.mulRight₀ (n/j) (div_ne_zero hn hj)) |
| have he (t : ℝ) : e' t = (n*t-j)/(j/n) := by |
| dsimp [e',e] |
| field_simp |
| ring |
| have hc' := hh.comp_homeomorph e' |
| have heq : unfoldedModeDensity I J (invFunOn J I) (deriv J) q n j = |
| fun t => |j/n|⁻¹ * branchDensity I J (invFunOn J I) (deriv J) (fun x => n*q x) (e' t) := by |
| funext t |
| rw [he] |
| rfl |
| rw [heq] |
| exact hc'.mul_left |
|
|
| lemma periodizedReal_eq_periodize {q : ℝ → ℝ} (hc : HasCompactSupport q) : |
| periodizedReal q = periodize q := by |
| funext x |
| obtain ⟨R,hR⟩ := hc.isBounded.exists_norm_le |
| obtain ⟨N,hN⟩ := exists_nat_gt (R+|x|+1) |
| have hs : support (fun k : ℤ => q (x+k)) ⊆ (Finset.Icc (-(N:ℤ)) N : Set ℤ) := by |
| intro k hk |
| have hr := hR (x+k) (subset_tsupport q hk) |
| have hh : |(k:ℝ)| < N := by |
| have ha := abs_sub_le (x+(k:ℝ)) 0 x |
| simp only [sub_zero,zero_sub,abs_neg,add_sub_cancel_left,Real.norm_eq_abs] at hr ha |
| linarith |
| have hab := abs_lt.mp hh |
| simp only [Finset.mem_coe,Finset.mem_Icc] |
| constructor |
| · exact_mod_cast (by linarith : -(N:ℝ) ≤ (k:ℝ)) |
| · exact_mod_cast (by linarith : (k:ℝ) ≤ (N:ℝ)) |
| exact tsum_eq_finsum ((Finset.finite_toSet _).subset hs) |
|
|
| lemma foldCircleDensity_smooth_lift {q : ℝ → ℝ} |
| (hq : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) q) (hc : HasCompactSupport q) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun x : ℝ => foldCircleDensity q (x:UnitAddCircle)) := by |
| have he : (fun x : ℝ => foldCircleDensity q (x:UnitAddCircle)) = periodize q := by |
| funext x |
| rw [foldCircleDensity_coe_all] |
| exact congrFun (periodizedReal_eq_periodize hc) x |
| rw [he] |
| exact contDiff_periodize hq hc |
|
|
| / |
| is taken before the final passage integer, as in the source. -/ |
| def realRetainedWeight (ψ₀ : ℝ → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) (x : ℝ) : ℝ := |
| ψ₀ x^2*χ ((j:ℝ)/n) x^2*‖exactPassageCoeff m (H n) A j x‖^2 |
|
|
| lemma realRetainedWeight_support (ψ₀ : ℝ → ℝ) (χ : ℝ → UnitAddCircle → ℝ) |
| (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) : |
| tsupport (realRetainedWeight ψ₀ χ m H A n j) ⊆ tsupport ψ₀ := by |
| apply closure_mono |
| intro x hx hz |
| exact hx (by simp [realRetainedWeight,hz]) |
|
|
| lemma realRetainedWeight_compact {ψ₀ : ℝ → ℝ} (hc : HasCompactSupport ψ₀) |
| (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) : |
| HasCompactSupport (realRetainedWeight ψ₀ χ m H A n j) := |
| hc.of_isClosed_subset isClosed_closure (realRetainedWeight_support ψ₀ χ m H A n j) |
|
|
| lemma realRetainedWeight_smooth {ψ₀ : ℝ → ℝ} {χ : ℝ → UnitAddCircle → ℝ} |
| {m : ℤ} {H : ℕ → UnitAddCircle → UnitAddCircle} {Hl : ℕ → ℝ → ℝ} |
| {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ψ₀) |
| (hχ : CylinderSmooth χ.uncurry) |
| (hHl : ∀ n, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n)) |
| (hH : ∀ n (t : ℝ), H n (t:UnitAddCircle) = (Hl n t:UnitAddCircle)) |
| (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) |
| (n : ℕ) (j : ℤ) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (realRetainedWeight ψ₀ χ m H A n j) := by |
| have hχ' : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun x : ℝ => χ ((j:ℝ)/n) x) := hχ.comp (contDiff_const.prodMk contDiff_id) |
| exact ((hψ.pow 2).mul (hχ'.pow 2)).mul |
| ((smooth_exactPassageCoeff (hHl n) (hH n) hA j).norm_sq (𝕜 := ℂ)) |
|
|
| lemma retainedWeight_lift_eq {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} |
| (χ : ℝ → UnitAddCircle → ℝ) (m : ℤ) (H : ℕ → UnitAddCircle → UnitAddCircle) |
| (A : UnitAddCircle × UnitAddCircle → ℂ) (n : ℕ) (j : ℤ) |
| (he : ∀ x ∈ Ioc (0:ℝ) (0+1), ψ (x:UnitAddCircle) = ψ₀ x) |
| (hs : tsupport ψ₀ ⊆ Ioc (0:ℝ) (0+1)) : |
| circleDensityLift 0 (retainedCircleWeight ψ χ |
| (fun n j => exactPassageCoeff m (H n) A j) n j) = |
| realRetainedWeight ψ₀ χ m H A n j := by |
| apply circleDensityLift_eq |
| · intro x hx |
| simp only [retainedCircleWeight,realRetainedWeight,he x hx] |
| · exact (subset_tsupport _).trans ((realRetainedWeight_support ψ₀ χ m H A n j).trans hs) |
|
|
| / |
| zero set. This supplies the next passage's regularity; it is not an assumption |
| that the density is already smooth. -/ |
| theorem exactRetainedPacketDensity_smooth_lift |
| {I : Set ℝ} {J : ℝ → ℝ} {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} |
| {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} |
| {H : ℕ → UnitAddCircle → UnitAddCircle} {Hl : ℕ → ℝ → ℝ} |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {S : Finset ℤ} {n : ℕ} |
| (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J) (hIo : IsOpen I) |
| (hinj : InjOn J I) (hd : ∀ x ∈ I, deriv J x ≠ 0) |
| (hψ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ψ₀) (hc : HasCompactSupport ψ₀) |
| (he : ∀ x ∈ Ioc (0:ℝ) (0+1), ψ (x:UnitAddCircle) = ψ₀ x) |
| (hs : tsupport ψ₀ ⊆ Ioc (0:ℝ) (0+1) ∩ I) |
| (hχ : CylinderSmooth χ.uncurry) |
| (hHl : ∀ n, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (Hl n)) |
| (hH : ∀ n (t : ℝ), H n (t:UnitAddCircle) = (Hl n t:UnitAddCircle)) |
| (hA : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) |
| (hn : 0 < n) (hS : ∀ j ∈ S, j ≠ 0) : |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun x : ℝ => exactRetainedPacketDensity I J ψ χ m H A S n x) := by |
| unfold exactRetainedPacketDensity |
| simp_rw [retainedWeight_lift_eq χ m H A n _ he (hs.trans inter_subset_left)] |
| apply foldCircleDensity_smooth_lift |
| · apply ContDiff.sum |
| intro j hj |
| exact unfoldedModeDensity_smooth hJ hIo hinj hd |
| (realRetainedWeight_smooth hψ hχ hHl hH hA n j) |
| (realRetainedWeight_compact hc χ m H A n j) |
| ((realRetainedWeight_support ψ₀ χ m H A n j).trans (hs.trans inter_subset_right)) n j |
| · have hh (j : ℤ) (hj : j ∈ S) : HasCompactSupport |
| (unfoldedModeDensity I J (invFunOn J I) (deriv J) |
| (realRetainedWeight ψ₀ χ m H A n j) n j) := |
| unfoldedModeDensity_compact hJ.continuous |
| (realRetainedWeight_compact hc χ m H A n j) |
| (by exact_mod_cast hn.ne') (by exact_mod_cast hS j hj) |
| convert HasCompactSupport.finset_sum hh using 1 |
| first | rfl | (funext time; simp only [Finset.sum_apply]) |
|
|
| lemma continuous_of_continuous_lift {Y : Type*} [TopologicalSpace Y] |
| {f : UnitAddCircle → Y} (hf : Continuous (fun x : ℝ => f (x:UnitAddCircle))) : |
| Continuous f := |
| isQuotientMap_quotient_mk'.continuous_iff.mpr hf |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! Identification of the reference stationary marginal with the actual old |
| packet spectrum. In particular integer carrier speeds do not change it. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators |
| namespace ThreeTorus |
|
|
| lemma positive_circle_branch_fold |
| {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} |
| {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} |
| (hq : Integrable q AddCircle.haarAddCircle) (hqm : Measurable q₀) |
| (hq0 : ∀ a, 0 ≤ q a) (hq00 : ∀ x, 0 ≤ q₀ x) |
| (hq₀ : ∀ x ∈ Ioc o (o+1), q (x:UnitAddCircle) = q₀ x) |
| (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) |
| (hj : Measurable J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) |
| (hJe : ∀ x ∈ I, J (x:UnitAddCircle) = J₀ x) |
| (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) : |
| (AddCircle.haarAddCircle.withDensity (fun a => ENNReal.ofReal (q a))).map |
| (fun a => (J a:UnitAddCircle)) = |
| AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal |
| (foldCircleDensity (branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀) z)) := by |
| let w := branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ |
| have hqi : Integrable q₀ := by |
| rw [← circleDensityLift_eq hq₀ hq₀s] |
| exact integrable_circleDensityLift hq o |
| have hD : ∀ x ∈ I, HasDerivWithinAt J₀ (deriv J₀ x) I x := |
| fun x _ => (hJ.differentiable (by simp) x).hasDerivAt.hasDerivWithinAt |
| have hIm := (isOpen_image_of_deriv_ne_zero hJ hIo hd).measurableSet |
| have hwi : Integrable w := integrable_branchDensity hIo.measurableSet hIm hD hinj |
| (fun _ hx => hinj.leftInvOn_invFunOn hx) hd hqi.integrableOn |
| have hwm : Measurable w := measurable_branchDensity_smooth hJ hIo hinj hd hqm |
| have hmap := circle_branch_pushforward hq hq₀ hq₀s hqs hj hJ.continuous.measurable hJe |
| hIo.measurableSet hIm hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd |
| apply positive_density_map_of_vectorMeasure (AddCircle.measurable_mk'.comp hj) hq |
| (integrable_foldCircleDensity hwm hwi) hq0 |
| (foldCircleDensity_nonneg (branchDensity_nonneg hq00)) |
| rw [← foldCircleDensity_pushforward hwm hwi,← hmap] |
| exact (vectorMeasure_map_comp _ hj AddCircle.measurable_mk').symm |
|
|
| lemma stationaryWeight_change_branch {A : ℝ → ℂ} {J : ℝ → ℝ} {C I : Set ℝ} |
| (hIC : I ⊆ C) (hi : InjOn J C) (hs : support A ⊆ I) : |
| stationaryWeight C J A = stationaryWeight I J A := by |
| have h := stationaryWeight_family_of_branch (Ω := Unit) hIC hi (fun _ => hs) |
| funext s |
| have hh := congrFun h (s,()) |
| exact hh |
|
|
| / |
| model. The integer speed carrier vanishes on the temporal circle. -/ |
| theorem oldPacket_spectralDensity_of_modulus |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : ℝ → ℂ} |
| {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m c : ℤ} {o : ℝ} {C I : Set ℝ} |
| (hA : Continuous A) (hA₀ : Continuous A₀) |
| (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) |
| (hAe : ∀ x ∈ Ioc o (o+1), A ((x:UnitAddCircle),0) = A₀ x) |
| (has : support A₀ ⊆ Ioc o (o+1)) (hAs : support A₀ ⊆ I) |
| (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) |
| (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) |
| (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) |
| (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) : |
| packetSpectralMeasure (fun a => ((v a-c:ℝ):UnitAddCircle)) m A = |
| AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal |
| (foldCircleDensity (stationaryWeight C (fun x => (m:ℝ)*V x) A₀) z)) := by |
| have hqc : Continuous (fun u : UnitAddCircle => ‖A (u,0)‖^2) := |
| ((hA.comp (continuous_id.prodMk continuous_const)).norm.pow 2) |
| have hps := packetSpectralMeasure_of_modulus |
| (φ := fun a => (((v a-(c:ℝ)) : ℝ):UnitAddCircle)) (j := m) (B := A) |
| (q := fun u => ‖A (u,0)‖^2) |
| (AddCircle.measurable_mk'.comp (hv.sub measurable_const)) hqc.measurable |
| (fun p => by rw [hmod]) |
| rw [hps] |
| have hmode (a : UnitAddCircle) : m • ((v a-c:ℝ):UnitAddCircle) = ((m:ℝ)*v a:UnitAddCircle) := by |
| rw [← AddCircle.coe_zsmul] |
| simp only [zsmul_eq_mul] |
| rw [mul_sub] |
| have he : ((m:ℝ)*(c:ℝ):UnitAddCircle) = 0 := by |
| rw [← Int.cast_mul] |
| exact (AddCircle.coe_eq_zero_iff 1).mpr ⟨m*c,by simp⟩ |
| rw [AddCircle.coe_sub,he,sub_zero] |
| simp_rw [hmode] |
| rw [stationaryWeight_change_branch hIC hinj hAs] |
| apply positive_circle_branch_fold (q₀ := fun x => ‖A₀ x‖^2) |
| (hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| ((hA₀.norm.pow 2).measurable) (fun _ => sq_nonneg _) (fun _ => sq_nonneg _) |
| (fun x hx => by rw [hAe x hx]) |
| (by intro x hx; apply has; intro hz; apply hx; simp [hz]) |
| (by intro x hx; apply hAs; intro hz; apply hx; simp [hz]) |
| (hv.const_mul (m:ℝ)) (contDiff_const.mul hV) (fun x hx => by rw [hVe x hx]) |
| hIo (hinj.mono hIC) hd |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! Actual old densities for amplitudes depending on both slow coordinates. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators ENNReal |
| namespace ThreeTorus |
|
|
| lemma map_fst_withDensity_marginal |
| {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] |
| {μ : Measure X} {ν : Measure Y} [SFinite μ] [SFinite ν] |
| {q : X × Y → ℝ≥0∞} (hq : Measurable q) : |
| ((μ.prod ν).withDensity q).map Prod.fst = |
| μ.withDensity (fun x => ∫⁻ y, q (x,y) ∂ν) := by |
| ext s hs |
| rw [Measure.map_apply measurable_fst hs,withDensity_apply _ (measurable_fst hs), |
| withDensity_apply _ hs,← lintegral_indicator (measurable_fst hs), |
| ← lintegral_indicator hs,lintegral_prod _ (hq.indicator (measurable_fst hs)).aemeasurable] |
| apply lintegral_congr |
| intro x |
| by_cases hx : x ∈ s |
| · simp only [Set.indicator_of_mem hx] |
| apply lintegral_congr |
| intro y |
| exact Set.indicator_of_mem (show (x,y) ∈ Prod.fst ⁻¹' s from hx) q |
| · simp only [Set.indicator_of_notMem hx] |
| have hz (y : Y) : ((Prod.fst ⁻¹' s).indicator q) (x,y) = 0 := |
| Set.indicator_of_notMem (show (x,y) ∉ Prod.fst ⁻¹' s from hx) q |
| simp only [hz,lintegral_zero] |
|
|
| lemma continuous_slowMarginal {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hA : Continuous A) : |
| Continuous (fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle) := by |
| have hc : Continuous (uncurry (fun (u a : UnitAddCircle) => ‖A (u,a)‖^2)) := hA.norm.pow 2 |
| simpa only [setIntegral_univ] using |
| continuous_parametric_integral_of_continuous (X := UnitAddCircle) (Y := UnitAddCircle) |
| (μ := AddCircle.haarAddCircle) hc isCompact_univ |
|
|
| lemma packetSpectralMeasure_marginal {φ : UnitAddCircle → UnitAddCircle} |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {m : ℤ} |
| (hφ : Measurable φ) (hA : Continuous A) : |
| packetSpectralMeasure φ m A = |
| (AddCircle.haarAddCircle.withDensity (fun u => ENNReal.ofReal |
| (∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle))).map (fun u => m • φ u) := by |
| unfold packetSpectralMeasure slowVolume |
| change Measure.map ((fun u => m • φ u) ∘ Prod.fst) _ = _ |
| rw [← Measure.map_map (show Measurable (fun u => m • φ u) from hφ.const_smul m) measurable_fst, |
| map_fst_withDensity_marginal (q := fun p => ENNReal.ofReal (‖A p‖^2)) |
| (by simpa only [Pi.pow_apply] using (hA.norm.pow 2).measurable.ennreal_ofReal)] |
| congr 2 |
| funext u |
| exact (ofReal_integral_eq_lintegral_ofReal |
| (((hA.comp (continuous_const.prodMk continuous_id)).norm.pow 2).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _)) (ae_of_all _ (fun a => sq_nonneg _))).symm |
|
|
| lemma branchDensity_integral {Ω : Type*} [MeasurableSpace Ω] {ν : Measure Ω} |
| (I : Set ℝ) (J g D : ℝ → ℝ) (q : ℝ × Ω → ℝ) (s : ℝ) : |
| branchDensity I J g D (fun u => ∫ a, q (u,a) ∂ν) s = |
| ∫ a, branchDensity I J g D (fun u => q (u,a)) s ∂ν := by |
| classical |
| unfold branchDensity |
| by_cases hs : s ∈ J '' I |
| · simp only [Set.indicator_of_mem hs,integral_div] |
| · simp only [Set.indicator_of_notMem hs,integral_zero] |
|
|
| / |
| its temporal density. Spare-coordinate independence is not required. -/ |
| theorem oldPacket_spectralDensity |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} |
| {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m c : ℤ} {o : ℝ} {C I : Set ℝ} |
| (hA : Continuous A) (hA₀ : Continuous A₀.uncurry) |
| (hAe : ∀ a x, x ∈ Ioc o (o+1) → A ((x:UnitAddCircle),a) = A₀ a x) |
| (has : ∀ a, support (A₀ a) ⊆ Ioc o (o+1)) (hAs : ∀ a, support (A₀ a) ⊆ I) |
| (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) |
| (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) |
| (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) |
| (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) : |
| packetSpectralMeasure (fun a => ((v a-c:ℝ):UnitAddCircle)) m A = |
| AddCircle.haarAddCircle.withDensity (fun z => ENNReal.ofReal |
| (foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) |
| (A₀ a) s ∂AddCircle.haarAddCircle) z)) := by |
| let q : UnitAddCircle → ℝ := fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle |
| let q₀ : ℝ → ℝ := fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle |
| have hqc : Continuous q := continuous_slowMarginal hA |
| have hq₀c : Continuous q₀ := by |
| have hc : Continuous (uncurry (fun (x : ℝ) (a : UnitAddCircle) => ‖A₀ a x‖^2)) := |
| (hA₀.comp continuous_swap).norm.pow 2 |
| simpa only [q₀,setIntegral_univ] using |
| continuous_parametric_integral_of_continuous (X := ℝ) (Y := UnitAddCircle) |
| (μ := AddCircle.haarAddCircle) hc isCompact_univ |
| have hs (K : Set ℝ) (hK : ∀ a, support (A₀ a) ⊆ K) : support q₀ ⊆ K := by |
| intro x hx |
| by_contra hnot |
| apply hx |
| have hz (a : UnitAddCircle) : A₀ a x = 0 := by |
| by_contra hn |
| exact hnot (hK a hn) |
| simp only [q₀,hz,norm_zero,zero_pow (by decide : 2 ≠ 0),integral_zero] |
| have hmap := packetSpectralMeasure_marginal (m := m) (A := A) |
| (φ := fun a => (((v a-(c:ℝ)) : ℝ):UnitAddCircle)) |
| (AddCircle.measurable_mk'.comp (hv.sub measurable_const)) hA |
| rw [hmap] |
| have hmode (a : UnitAddCircle) : m • ((v a-c:ℝ):UnitAddCircle) = |
| (((m:ℝ)*v a:ℝ):UnitAddCircle) := by |
| rw [← AddCircle.coe_zsmul] |
| simp only [zsmul_eq_mul] |
| rw [mul_sub] |
| have he : ((m:ℝ)*(c:ℝ):UnitAddCircle) = 0 := by |
| rw [← Int.cast_mul] |
| exact (AddCircle.coe_eq_zero_iff 1).mpr ⟨m*c,by simp⟩ |
| rw [AddCircle.coe_sub,he,sub_zero] |
| simp_rw [hmode] |
| have hb : branchDensity I (fun x => (m:ℝ)*V x) |
| (invFunOn (fun x => (m:ℝ)*V x) I) (deriv (fun x => (m:ℝ)*V x)) q₀ = |
| fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s |
| ∂AddCircle.haarAddCircle := by |
| funext s |
| rw [show q₀ = fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle from rfl, |
| branchDensity_integral (q := fun p : ℝ × UnitAddCircle => ‖A₀ p.2 p.1‖^2)] |
| apply integral_congr_ae |
| filter_upwards [] with a |
| exact congrFun (stationaryWeight_change_branch hIC hinj (hAs a)).symm s |
| rw [← hb] |
| exact positive_circle_branch_fold |
| (hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| hq₀c.measurable (fun _ => integral_nonneg (fun _ => sq_nonneg _)) |
| (fun _ => integral_nonneg (fun _ => sq_nonneg _)) |
| (fun x hx => integral_congr_ae (ae_of_all _ (fun a => by dsimp only; rw [hAe a x hx]))) |
| (hs _ has) (hs _ hAs) (hv.const_mul (m:ℝ)) (contDiff_const.mul hV) |
| (fun x hx => by rw [hVe x hx]) hIo (hinj.mono hIC) hd |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! The unchanged old spectral marginal, as a signed pushforward, for applying |
| L¹ contraction to physical cutoffs. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators ENNReal |
| namespace ThreeTorus |
|
|
| lemma signed_circle_branch_fold |
| {q : UnitAddCircle → ℝ} {q₀ J₀ : ℝ → ℝ} |
| {J : UnitAddCircle → ℝ} {o : ℝ} {I : Set ℝ} |
| (hq : Integrable q AddCircle.haarAddCircle) (hqm : Measurable q₀) |
| (hq₀ : ∀ x ∈ Ioc o (o+1), q (x:UnitAddCircle) = q₀ x) |
| (hq₀s : support q₀ ⊆ Ioc o (o+1)) (hqs : support q₀ ⊆ I) |
| (hj : Measurable J) (hJ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) J₀) |
| (hJe : ∀ x ∈ I, J (x:UnitAddCircle) = J₀ x) |
| (hIo : IsOpen I) (hinj : InjOn J₀ I) (hd : ∀ x ∈ I, deriv J₀ x ≠ 0) : |
| Integrable (foldCircleDensity (branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀)) |
| AddCircle.haarAddCircle ∧ |
| (AddCircle.haarAddCircle.withDensityᵥ q).map (fun a => (J a:UnitAddCircle)) = |
| AddCircle.haarAddCircle.withDensityᵥ |
| (foldCircleDensity (branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀)) := by |
| let w := branchDensity I J₀ (invFunOn J₀ I) (deriv J₀) q₀ |
| have hqi : Integrable q₀ := by |
| rw [← circleDensityLift_eq hq₀ hq₀s] |
| exact integrable_circleDensityLift hq o |
| have hD : ∀ x ∈ I, HasDerivWithinAt J₀ (deriv J₀ x) I x := |
| fun x _ => (hJ.differentiable (by simp) x).hasDerivAt.hasDerivWithinAt |
| have hIm := (isOpen_image_of_deriv_ne_zero hJ hIo hd).measurableSet |
| have hwi : Integrable w := integrable_branchDensity hIo.measurableSet hIm hD hinj |
| (fun _ hx => hinj.leftInvOn_invFunOn hx) hd hqi.integrableOn |
| have hwm : Measurable w := measurable_branchDensity_smooth hJ hIo hinj hd hqm |
| have hmap := circle_branch_pushforward hq hq₀ hq₀s hqs hj hJ.continuous.measurable hJe |
| hIo.measurableSet hIm hD hinj (fun _ hx => hinj.leftInvOn_invFunOn hx) hd |
| refine ⟨integrable_foldCircleDensity hwm hwi,?_⟩ |
| rw [← foldCircleDensity_pushforward hwm hwi,← hmap] |
| exact (vectorMeasure_map_comp _ hj AddCircle.measurable_mk').symm |
|
|
| theorem oldStationaryMarginal_map |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} |
| {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m : ℤ} {o : ℝ} {C I : Set ℝ} |
| (hA : Continuous A) (hA₀ : Continuous A₀.uncurry) |
| (hAe : ∀ a x, x ∈ Ioc o (o+1) → A ((x:UnitAddCircle),a) = A₀ a x) |
| (has : ∀ a, support (A₀ a) ⊆ Ioc o (o+1)) (hAs : ∀ a, support (A₀ a) ⊆ I) |
| (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) |
| (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) |
| (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) |
| (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) : |
| let p := foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) |
| (A₀ a) s ∂AddCircle.haarAddCircle) |
| Integrable p AddCircle.haarAddCircle ∧ |
| (AddCircle.haarAddCircle.withDensityᵥ (fun u => ∫ a, ‖A (u,a)‖^2 |
| ∂AddCircle.haarAddCircle)).map (fun u => (((m:ℝ)*v u : ℝ):UnitAddCircle)) = |
| AddCircle.haarAddCircle.withDensityᵥ p := by |
| dsimp only |
| let q : UnitAddCircle → ℝ := fun u => ∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle |
| let q₀ : ℝ → ℝ := fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle |
| have hqc : Continuous q := continuous_slowMarginal hA |
| have hq₀c : Continuous q₀ := by |
| have hc : Continuous (uncurry (fun (x : ℝ) (a : UnitAddCircle) => ‖A₀ a x‖^2)) := |
| (hA₀.comp continuous_swap).norm.pow 2 |
| simpa only [q₀,setIntegral_univ] using |
| continuous_parametric_integral_of_continuous (X := ℝ) (Y := UnitAddCircle) |
| (μ := AddCircle.haarAddCircle) hc isCompact_univ |
| have hs (K : Set ℝ) (hK : ∀ a, support (A₀ a) ⊆ K) : support q₀ ⊆ K := by |
| intro x hx |
| by_contra hnot |
| apply hx |
| have hz (a : UnitAddCircle) : A₀ a x = 0 := by |
| by_contra hn |
| exact hnot (hK a hn) |
| simp only [q₀,hz,norm_zero,zero_pow (by decide : 2 ≠ 0),integral_zero] |
| have hb : branchDensity I (fun x => (m:ℝ)*V x) |
| (invFunOn (fun x => (m:ℝ)*V x) I) (deriv (fun x => (m:ℝ)*V x)) q₀ = |
| fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) (A₀ a) s |
| ∂AddCircle.haarAddCircle := by |
| funext s |
| rw [show q₀ = fun x => ∫ a, ‖A₀ a x‖^2 ∂AddCircle.haarAddCircle from rfl, |
| branchDensity_integral (q := fun p : ℝ × UnitAddCircle => ‖A₀ p.2 p.1‖^2)] |
| apply integral_congr_ae |
| filter_upwards [] with a |
| exact congrFun (stationaryWeight_change_branch hIC hinj (hAs a)).symm s |
| rw [← hb] |
| exact signed_circle_branch_fold |
| (hqc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| hq₀c.measurable |
| (fun x hx => integral_congr_ae (ae_of_all _ (fun a => by dsimp only; rw [hAe a x hx]))) |
| (hs _ has) (hs _ hAs) (hv.const_mul (m:ℝ)) (contDiff_const.mul hV) |
| (fun x hx => by rw [hVe x hx]) hIo (hinj.mono hIC) hd |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! The neutral passage's old cutoff reference differs from the actual old |
| packet density by at most its physical missing mass. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators |
| namespace ThreeTorus |
|
|
| lemma cutoff_slowMarginal_l1 |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {ψ : UnitAddCircle → ℝ} |
| (hA : Continuous A) (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) |
| {B : ℝ} (_hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) : |
| (∫ u, |(∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle)- |
| (∫ a, ‖(ψ a:ℂ)*A (u,a)‖^2 ∂AddCircle.haarAddCircle)| ∂AddCircle.haarAddCircle) ≤ |
| B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) := by |
| let D := B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) |
| have hd : ∀ u, |(∫ a, ‖A (u,a)‖^2 ∂AddCircle.haarAddCircle)- |
| (∫ a, ‖(ψ a:ℂ)*A (u,a)‖^2 ∂AddCircle.haarAddCircle)| ≤ D := by |
| intro u |
| have ha : Continuous (fun a => ‖A (u,a)‖^2) := |
| (hA.comp (continuous_const.prodMk continuous_id)).norm.pow 2 |
| have hap : Continuous (fun a => ‖(ψ a:ℂ)*A (u,a)‖^2) := |
| ((Complex.continuous_ofReal.comp hψc).mul |
| (hA.comp (continuous_const.prodMk continuous_id))).norm.pow 2 |
| have hc : Continuous (fun a => ‖A (u,a)‖^2-‖(ψ a:ℂ)*A (u,a)‖^2) := ha.sub hap |
| have he (a : UnitAddCircle) : ‖A (u,a)‖^2-‖(ψ a:ℂ)*A (u,a)‖^2 = |
| (1-ψ a^2)*‖A (u,a)‖^2 := by |
| simp only [norm_mul,mul_pow,Complex.norm_real,Real.norm_eq_abs,sq_abs] |
| ring |
| have hz (a : UnitAddCircle) : 0 ≤ ‖A (u,a)‖^2-‖(ψ a:ℂ)*A (u,a)‖^2 := by |
| rw [he] |
| exact mul_nonneg (by nlinarith [(hψ a).1,(hψ a).2]) (sq_nonneg _) |
| rw [← integral_sub (ha.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| (hap.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)), |
| abs_of_nonneg (integral_nonneg hz)] |
| have hbound : Continuous (fun a => B*(1-ψ a^2)) := |
| continuous_const.mul (continuous_const.sub (hψc.pow 2)) |
| calc |
| _ ≤ ∫ a, B*(1-ψ a^2) ∂AddCircle.haarAddCircle := by |
| apply integral_mono (hc.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| (hbound.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| intro a |
| dsimp only |
| rw [he,mul_comm B] |
| exact mul_le_mul_of_nonneg_left (hAb _) (by nlinarith [(hψ a).1,(hψ a).2]) |
| _ = D := by rw [integral_const_mul] |
| calc |
| _ ≤ ∫ _ : UnitAddCircle, D ∂AddCircle.haarAddCircle := by |
| exact integral_mono_of_nonneg (ae_of_all _ (fun _ => abs_nonneg _)) (integrable_const D) |
| (ae_of_all _ hd) |
| _ = _ := by simp [D] |
|
|
| / |
| old spectrum. Its loss is bounded independently of slope, residue modulus or |
| passage integer. -/ |
| theorem oldStationaryMarginal_cutoff_l1 |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} |
| {v : UnitAddCircle → ℝ} {V : ℝ → ℝ} {m : ℤ} {o : ℝ} {C I : Set ℝ} |
| (hA : Continuous A) (hA₀ : Continuous A₀.uncurry) |
| (hAe : ∀ a x, x ∈ Ioc o (o+1) → A ((x:UnitAddCircle),a) = A₀ a x) |
| (has : ∀ a, support (A₀ a) ⊆ Ioc o (o+1)) (hAs : ∀ a, support (A₀ a) ⊆ I) |
| (hv : Measurable v) (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) |
| (hVe : ∀ x ∈ I, v (x:UnitAddCircle) = V x) |
| (hIo : IsOpen I) (hIC : I ⊆ C) (hinj : InjOn (fun x => (m:ℝ)*V x) C) |
| (hd : ∀ x ∈ I, deriv (fun x => (m:ℝ)*V x) x ≠ 0) |
| {ψ : UnitAddCircle → ℝ} (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) |
| {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) : |
| let p := foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) |
| (A₀ a) s ∂AddCircle.haarAddCircle) |
| let pψ := foldCircleDensity (fun s => ∫ a, stationaryWeight C (fun x => (m:ℝ)*V x) |
| (fun x => (ψ a:ℂ)*A₀ a x) s ∂AddCircle.haarAddCircle) |
| (∫ z, |pψ z-p z| ∂AddCircle.haarAddCircle) ≤ |
| B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) := by |
| dsimp only |
| let Aψ : UnitAddCircle × UnitAddCircle → ℂ := fun p => (ψ p.2:ℂ)*A p |
| let A₀ψ : UnitAddCircle → ℝ → ℂ := fun a x => (ψ a:ℂ)*A₀ a x |
| have hAψ : Continuous Aψ := ((Complex.continuous_ofReal.comp hψc).comp continuous_snd).mul hA |
| have hA₀ψ : Continuous A₀ψ.uncurry := |
| ((Complex.continuous_ofReal.comp hψc).comp continuous_fst).mul hA₀ |
| have hs (K : Set ℝ) (hK : ∀ a, support (A₀ a) ⊆ K) : ∀ a, support (A₀ψ a) ⊆ K := by |
| intro a x hx |
| exact hK a (right_ne_zero_of_mul hx) |
| have he : ∀ a x, x ∈ Ioc o (o+1) → Aψ ((x:UnitAddCircle),a) = A₀ψ a x := by |
| intro a x hx |
| dsimp only [Aψ,A₀ψ] |
| rw [hAe a x hx] |
| obtain ⟨hpi,hmap⟩ := oldStationaryMarginal_map hA hA₀ hAe has hAs hv hV hVe hIo hIC hinj hd |
| obtain ⟨hψi,hψmap⟩ := oldStationaryMarginal_map hAψ hA₀ψ he (hs _ has) (hs _ hAs) |
| hv hV hVe hIo hIC hinj hd |
| have hq := continuous_slowMarginal hA |
| have hqψ := continuous_slowMarginal hAψ |
| have hb := l1_pushforward_difference |
| (hqψ.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) |
| (hq.integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _)) hψi hpi |
| (AddCircle.measurable_mk'.comp (hv.const_mul (m:ℝ))) hψmap hmap |
| refine hb.trans ?_ |
| simp only [Aψ,abs_sub_comm] |
| exact cutoff_slowMarginal_l1 hA hψc hψ hB hAb |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! Quantitative physical and spectral conclusions for the same actual packet. |
| This is the analytic (not yet geometric finite-stage) transmission boundary. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators FourierTransform |
| namespace ThreeTorus |
|
|
| lemma sq_norm_sub_le_two (x y z : ℂ) : |
| ‖x-z‖^2 ≤ 2*‖x-y‖^2+2*‖y-z‖^2 := by |
| have hh : ‖x-z‖ ≤ ‖x-y‖+‖y-z‖ := by |
| convert norm_add_le (x-y) (y-z) using 1 |
| congr 1 |
| ring |
| have h0 := norm_nonneg (x-z) |
| have h1 := norm_nonneg (x-y) |
| have h2 := norm_nonneg (y-z) |
| nlinarith [sq_nonneg (‖x-y‖-‖y-z‖)] |
|
|
| lemma continuous_passageChart {H : UnitAddCircle → UnitAddCircle} (hH : Continuous H) : |
| Continuous (passageChart H) := by |
| change Continuous (fun x : TripleTorus => (x.2.1,x.2.2,x.1+(-H x.2.1))) |
| exact continuous_snd.fst.prodMk (continuous_snd.snd.prodMk |
| (continuous_fst.add (hH.comp continuous_snd.fst).neg)) |
|
|
| lemma continuous_retainedPassage {ψ : UnitAddCircle → ℝ} |
| {χ : ℝ → UnitAddCircle → ℝ} {m : ℤ} |
| {H : ℕ → UnitAddCircle → UnitAddCircle} {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hψ : Continuous ψ) (hχ : Continuous χ.uncurry) |
| (hH : ∀ n, Continuous (H n)) (hA : Continuous A) (n : ℕ) (S : Finset ℤ) : |
| Continuous (retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) ψ) := by |
| change Continuous (fun x => ∑ j ∈ S, packet j (retainedAmplitude ψ χ m H A n j) x) |
| exact continuous_finsetSum _ (fun j _ => continuous_packet |
| (continuous_retainedAmplitude hψ hχ hH hA n j)) |
|
|
| lemma cutoff_packet_error_eq {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) (ψ : UnitAddCircle → ℝ) (m : ℤ) : |
| (∫ x : TripleTorus, ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) = |
| (∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)* |
| (∫ a, (1-ψ a)^2 ∂AddCircle.haarAddCircle) := by |
| have he (x : TripleTorus) : ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 = |
| ‖A (x.2.1,0)‖^2*(1-ψ x.2.2)^2 := by |
| rw [show packet m A x-(ψ x.2.2:ℂ)*packet m A x = |
| (1-(ψ x.2.2:ℂ))*packet m A x by ring] |
| simp only [packet,norm_mul,fourier_apply,Circle.norm_coe,mul_one] |
| rw [← Complex.ofReal_one,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs, |
| mul_pow,sq_abs,hmod] |
| ring |
| simp_rw [he] |
| change (∫ x : UnitAddCircle × (UnitAddCircle × UnitAddCircle), |
| ‖A (x.2.1,0)‖^2*(1-ψ x.2.2)^2 ∂AddCircle.haarAddCircle.prod slowVolume) = _ |
| rw [integral_fun_snd (μ := AddCircle.haarAddCircle) (ν := slowVolume) |
| (fun p : UnitAddCircle × UnitAddCircle => ‖A (p.1,0)‖^2*(1-ψ p.2)^2)] |
| simpa only [probReal_univ,one_smul] using |
| (integral_prod_mul (μ := AddCircle.haarAddCircle) (ν := AddCircle.haarAddCircle) |
| (fun u => ‖A (u,0)‖^2) (fun a => (1-ψ a)^2)) |
|
|
| lemma cutoff_packet_error_le {A : UnitAddCircle × UnitAddCircle → ℂ} |
| (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) {δ : ℝ} |
| (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (m : ℤ) : |
| (∫ x : TripleTorus, |
| ‖packet m A x-(neutralCircleCutoff c x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) ≤ |
| (∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)*(4*δ) := by |
| rw [cutoff_packet_error_eq hmod] |
| apply mul_le_mul_of_nonneg_left _ (integral_nonneg (fun _ => sq_nonneg _)) |
| have hc := continuous_neutralCircleCutoff c |
| have hi : Integrable (fun a => (1-neutralCircleCutoff c a^2)) AddCircle.haarAddCircle := |
| (continuous_const.sub (hc.pow 2)).integrable_of_hasCompactSupport |
| (HasCompactSupport.of_compactSpace _) |
| calc |
| _ ≤ ∫ a, (1-neutralCircleCutoff c a^2) ∂AddCircle.haarAddCircle := by |
| apply integral_mono_of_nonneg (ae_of_all _ (fun _ => sq_nonneg _)) hi |
| exact ae_of_all _ (fun a => by |
| have h := neutralCircleCutoff_bounds c hδ a |
| nlinarith) |
| _ ≤ _ := neutralCircleCutoff_error c hδ hδ' |
|
|
| / |
| Its missing mass is explicitly paid. -/ |
| lemma uncut_retainedPassage_error_le |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {m : ℤ} |
| {H : ℕ → UnitAddCircle → UnitAddCircle} {χ : ℝ → UnitAddCircle → ℝ} |
| (hA : Continuous A) (hH : ∀ n, Continuous (H n)) (hχ : Continuous χ.uncurry) |
| (hmod : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) {δ : ℝ} |
| (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (n : ℕ) (S : Finset ℤ) : |
| (∫ x : TripleTorus, |
| ‖packet m A x-retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) |
| (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) ≤ |
| 8*δ*(∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)+ |
| 2*(∫ x : TripleTorus, |
| ‖(neutralCircleCutoff c x.2.2:ℂ)*packet m A x- |
| retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) |
| (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) := by |
| let f := packet m A |
| let g := fun x : TripleTorus => (neutralCircleCutoff c x.2.2:ℂ)*f x |
| let h := fun x : TripleTorus => retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) |
| (neutralCircleCutoff c) (passageChart (H n) x) |
| have hf : Continuous f := continuous_packet hA |
| have hg : Continuous g := (Complex.continuous_ofReal.comp |
| ((continuous_neutralCircleCutoff c).comp continuous_snd.snd)).mul hf |
| have hh : Continuous h := |
| (continuous_retainedPassage (continuous_neutralCircleCutoff c) hχ hH hA n S).comp |
| (continuous_passageChart (hH n)) |
| have hi (v w : TripleTorus → ℂ) (hv : Continuous v) (hw : Continuous w) : |
| Integrable (fun x => ‖v x-w x‖^2) tripleVolume := |
| ((hv.sub hw).norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hfg := hi f g hf hg |
| have hgh := hi g h hg hh |
| calc |
| _ ≤ ∫ x, (2*‖f x-g x‖^2+2*‖g x-h x‖^2) ∂tripleVolume := by |
| apply integral_mono (hi f h hf hh) ((hfg.const_mul 2).add (hgh.const_mul 2)) |
| intro x |
| exact sq_norm_sub_le_two (f x) (g x) (h x) |
| _ = 2*(∫ x, ‖f x-g x‖^2 ∂tripleVolume)+2*(∫ x, ‖g x-h x‖^2 ∂tripleVolume) := by |
| rw [integral_add (hfg.const_mul 2) (hgh.const_mul 2),integral_const_mul,integral_const_mul] |
| _ ≤ _ := by |
| have hh := cutoff_packet_error_le hmod c hδ hδ' m |
| change (∫ x, ‖f x-g x‖^2 ∂tripleVolume) ≤ _ at hh |
| dsimp only [g,h] |
| nlinarith |
|
|
| / |
| spectral, smoothness and all bilateral-moment conclusions. The passage integer |
| remains arbitrarily large after the law, cutoffs and amplitudes are fixed. -/ |
| theorem exact_quantile_passage |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} |
| {Pr V : ℝ → ℝ} {m : ℤ} {Δ : ℕ → ℝ} {H : ℕ → UnitAddCircle → UnitAddCircle} |
| {o l u R : ℝ} |
| (hm : m ≠ 0) |
| (hH : ∀ n (x : ℝ), H n (x : UnitAddCircle) = ((n:ℝ)*(Pr x+Δ n*x) : UnitAddCircle)) |
| (hA : ∀ a x, x ∈ Ioc o (o+1) → A ((x : UnitAddCircle),a) = A₀ a x) |
| (hA₀ : ∀ a, ContDiff ℝ 2 (A₀ a)) (hAc : Continuous A₀.uncurry) |
| (hAd : Continuous (fun p : UnitAddCircle × ℝ => deriv (A₀ p.1) p.2)) |
| (hAr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun p : ℝ × ℝ => A₀ (p.2 : UnitAddCircle) p.1)) |
| (hsupp : ∀ a, support (A₀ a) ⊆ Ioc l u) |
| (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) |
| (hPr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Pr) |
| (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv Pr = V) |
| (hdV : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) |
| (h0 : 0 ∉ (fun x => (m:ℝ)*V x) '' Icc l u) |
| (hHc : ∀ n, Continuous (H n)) (hAcont : Continuous A) (hΔt : Tendsto Δ atTop (𝓝 0)) |
| (hmod : ∀ a x, ‖A₀ a x‖ = ‖A₀ 0 x‖) |
| (hmodA : ∀ u a, ‖A (u,a)‖ = ‖A (u,0)‖) |
| (hAs : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) |
| {ρ : ℝ → ℝ} (hρ : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) ρ) (hi : Integrable ρ) |
| (hp : ∀ y, 0 < ρ y) (hρm : ∫ y, ρ y = 1) |
| {C : ℝ} (hC : 0 ≤ C) (hb : ∀ y, |ρ y| ≤ C/(1+y^2)) {D : ℝ} |
| (hF : ∀ v, D < |v| → 𝓕 (fun y => (ρ y : ℂ)) v = 0) |
| {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) |
| {χ : ℝ → UnitAddCircle → ℝ} |
| (hχs : CylinderSmooth χ.uncurry) |
| (hχc : Continuous χ.uncurry) (hcχ : HasCompactSupport χ.uncurry) (hzχ : ∀ a, χ 0 a = 0) |
| (hχ : ∀ s a, 0 ≤ χ s a ∧ χ s a ≤ 1) |
| (hχW : ∀ s a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s ≠ 0 → χ s a = 1) |
| {g : UnitAddCircle → ℝ} (hg : Continuous g) {ε : ℝ} |
| (hideal : ∀ s ∈ support (stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ 0)), |
| ∀ y, |foldCircleDensity (scaledDensity ρ s) y-g y| ≤ ε) |
| {η : ℝ} (hη : 0 < η) : |
| let w := stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ 0) |
| ∃ S : ℕ → Finset ℤ, |
| (∀ n, 0 < n → ∀ j ∈ S n, j ≠ 0) ∧ |
| (∀ n, 0 < n → ∀ j ∉ S n, ∀ a, χ ((j:ℝ)/n) a = 0) ∧ |
| ∀ᶠ n : ℕ in atTop, |
| let ψ := neutralCircleCutoff c |
| let J := quantileCircleJitter ρ c |
| let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) |
| let f := retainedPassage m (H n) A (S n) (fun j => χ ((j:ℝ)/n)) ψ |
| let p := exactRetainedPacketDensity (Ioo δ (1-δ)) (lawJitter ρ c) ψ χ m H A (S n) n |
| (∫ x : TripleTorus, ‖packet m A x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < |
| 8*δ*(∫ u, ‖A (u,0)‖^2 ∂AddCircle.haarAddCircle)+η ∧ |
| (∫ z, |p z-foldCircleDensity w z*g ((n:ℤ) • z)| ∂AddCircle.haarAddCircle) < |
| (∫ s, w s)*(4*δ+ε)+η ∧ |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => p x) ∧ |
| (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter φ k x) ∂tripleVolume) = |
| ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ |
| (∀ j u a, ‖retainedAmplitude ψ χ m H A n j (u,a)‖ = |
| ‖retainedAmplitude ψ χ m H A n j (u,0)‖) := by |
| dsimp only |
| obtain ⟨S,hSN,hST,hsp⟩ := exact_quantile_circle_spectral_estimate hm hH hA hA₀ hAc hAd hAr |
| hsupp hlo huo hlu hR hPr hV hPV hdV h0 hHc hAcont hΔt hmod |
| hρ hi hp hρm hC hb hF c hδ hδ' hχc hcχ hzχ hχ hχW hg hideal hη |
| refine ⟨S,hSN,hST,?_⟩ |
| have hψc := continuous_neutralCircleCutoff c |
| have hph := exact_physical_retainedPassage_tendsto hm hH hA hA₀ hAc hAd hsupp |
| hlo huo hlu hR hPr hV hPV hdV hHc hAcont hΔt hχc hχ |
| (neutralCircleCutoff c) hψc (neutralCircleCutoff_bounds c hδ) hχW S hST |
| have hJs := lawJitter_smooth c hρ hi hp hρm |
| have hJinj := lawJitter_injOn c hρ.continuous hi hp hρm hδ |
| have hJd : ∀ x ∈ Ioo δ (1-δ), deriv (lawJitter ρ c) x ≠ 0 := by |
| intro x hx |
| obtain ⟨hd,hdp⟩ := lawJitter_deriv_pos c hρ hi hp hρm hδ hx |
| rw [hd.deriv] |
| exact hdp.ne' |
| have hψs : tsupport c.ψ ⊆ Ioc (0:ℝ) (0+1) ∩ Ioo δ (1-δ) := by |
| intro x hx |
| have hh := c.ψ_support hx |
| exact ⟨⟨by linarith [hh.1],by linarith [hh.2]⟩,hh⟩ |
| have heψ : ∀ x ∈ Ioc (0:ℝ) (0+1), neutralCircleCutoff c (x:UnitAddCircle) = c.ψ x := by |
| intro x hx |
| exact lawPsi_eq c hδ ⟨hx.1.le,by simpa using hx.2⟩ |
| have hψsupport : ∀ x ∈ Ioc (0:ℝ) 1, |
| neutralCircleCutoff c (x:UnitAddCircle) ≠ 0 → x ∈ Ioo δ (1-δ) := by |
| intro x hx hn |
| rw [heψ x (by simpa using hx)] at hn |
| exact (hψs (subset_tsupport c.ψ hn)).2 |
| have hHl : ∀ n : ℕ, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun x => (n:ℝ)*(Pr x+Δ n*x)) := |
| fun _ => contDiff_const.mul (hPr.add (contDiff_const.mul contDiff_id)) |
| filter_upwards [hsp,hph.eventually (eventually_lt_nhds (half_pos hη)), |
| eventually_gt_atTop (0:ℕ)] with n hnsp hnph hn |
| refine ⟨?_,hnsp,?_,?_,?_⟩ |
| · have hh := uncut_retainedPassage_error_le (m := m) hAcont hHc hχc hmodA c hδ hδ' n (S n) |
| exact hh.trans_lt (by linarith) |
| · exact exactRetainedPacketDensity_smooth_lift hJs isOpen_Ioo hJinj hJd |
| c.ψ_smooth c.ψ_compact heψ hψs hχs hHl hH hAs hn (hSN n hn) |
| · intro k |
| exact retainedPassage_spectral_moment hψc hχc hHc hAcont hψsupport |
| (continuous_quantileCircleJitter c hρ hi hp hρm) hJs |
| (fun x _ => quantileCircleJitter_coe ρ c x) isOpen_Ioo hJinj hJd hn (hSN n hn) k |
| · exact fun _ _ _ => retainedAmplitude_modulus _ _ _ _ _ _ _ _ _ |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! Quantitative good-arc cutoff losses for genuine two-coordinate amplitudes. |
| This does not presume spare-coordinate independence. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators |
| namespace ThreeTorus |
|
|
| lemma continuous_stationaryWeight_parameter {Ω : Type*} [TopologicalSpace Ω] |
| {A : Ω → ℝ → ℂ} (hA : Continuous A.uncurry) (C : Set ℝ) (J : ℝ → ℝ) (s : ℝ) : |
| Continuous (fun a => stationaryWeight C J (A a) s) := by |
| classical |
| unfold stationaryWeight branchDensity |
| by_cases hs : s ∈ J '' C |
| · simp only [Set.indicator_of_mem hs] |
| exact ((hA.comp (continuous_id.prodMk continuous_const)).norm.pow 2).div_const _ |
| · simp only [Set.indicator_of_notMem hs] |
| exact continuous_const |
|
|
| lemma stationaryCutoffLift_marginal {A : UnitAddCircle → ℝ → ℂ} |
| {ψ : UnitAddCircle → ℝ} {ψ₀ : ℝ → ℝ} {o : ℝ} |
| (heψ : ∀ x ∈ Ioc o (o+1), ψ₀ x = ψ (x:UnitAddCircle)) |
| (hsψ : support ψ₀ ⊆ Ioc o (o+1)) (C : Set ℝ) (J : ℝ → ℝ) (s : ℝ) : |
| (∫ a : ℝ, stationaryCutoffLift C J |
| (fun p : ℝ × ℝ => A (p.2:UnitAddCircle) p.1) ψ₀ (a,s)) = |
| ∫ a, stationaryWeight C J (fun x => (ψ a:ℂ)*A a x) s ∂AddCircle.haarAddCircle := by |
| let q : UnitAddCircle → ℝ := fun a => ψ a^2*stationaryWeight C J (A a) s |
| have he : circleDensityLift o q = fun a => stationaryCutoffLift C J |
| (fun p : ℝ × ℝ => A (p.2:UnitAddCircle) p.1) ψ₀ (a,s) := by |
| apply circleDensityLift_eq |
| · intro x hx |
| simp only [q,stationaryCutoffLift,heψ x hx] |
| · intro x hx |
| apply hsψ |
| intro hz |
| exact hx (by simp [stationaryCutoffLift,hz]) |
| rw [← he,circle_integral_lift] |
| apply integral_congr_ae |
| filter_upwards [] with a |
| rw [stationaryWeight_mul] |
| simp [q,Complex.norm_real,Real.norm_eq_abs,sq_abs] |
|
|
| lemma packet_cutoff_error_uniform {A : UnitAddCircle × UnitAddCircle → ℂ} |
| {ψ : UnitAddCircle → ℝ} (hψc : Continuous ψ) (hψ : ∀ a, 0 ≤ ψ a ∧ ψ a ≤ 1) |
| {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) (m : ℤ) : |
| (∫ x : TripleTorus, ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 ∂tripleVolume) ≤ |
| B*(∫ a, (1-ψ a^2) ∂AddCircle.haarAddCircle) := by |
| have hc : Continuous (fun x : TripleTorus => B*(1-ψ x.2.2^2)) := |
| continuous_const.mul (continuous_const.sub ((hψc.comp continuous_snd.snd).pow 2)) |
| have hi := hc.integrable_of_hasCompactSupport (μ := tripleVolume) |
| (HasCompactSupport.of_compactSpace _) |
| calc |
| _ ≤ ∫ x : TripleTorus, B*(1-ψ x.2.2^2) ∂tripleVolume := by |
| apply integral_mono_of_nonneg (ae_of_all _ (fun _ => sq_nonneg _)) hi |
| filter_upwards [] with x |
| have he : ‖packet m A x-(ψ x.2.2:ℂ)*packet m A x‖^2 = |
| ‖A (x.2.1,x.2.2)‖^2*(1-ψ x.2.2)^2 := by |
| rw [show packet m A x-(ψ x.2.2:ℂ)*packet m A x = |
| (1-(ψ x.2.2:ℂ))*packet m A x by ring] |
| simp only [packet,norm_mul,fourier_apply,Circle.norm_coe,mul_one] |
| rw [← Complex.ofReal_one,← Complex.ofReal_sub,Complex.norm_real,Real.norm_eq_abs, |
| mul_pow,sq_abs] |
| ring |
| rw [he] |
| have h := hψ x.2.2 |
| calc |
| _ ≤ B*(1-ψ x.2.2)^2 := mul_le_mul_of_nonneg_right (hAb _) (sq_nonneg _) |
| _ ≤ _ := mul_le_mul_of_nonneg_left (by nlinarith) hB |
| _ = _ := by |
| change (∫ x : UnitAddCircle × (UnitAddCircle × UnitAddCircle), |
| B*(1-ψ x.2.2^2) ∂AddCircle.haarAddCircle.prod slowVolume) = _ |
| rw [integral_fun_snd (μ := AddCircle.haarAddCircle (T := (1:ℝ))) (ν := slowVolume) |
| (fun p : UnitAddCircle × UnitAddCircle => B*(1-ψ p.2^2))] |
| simp only [probReal_univ,one_smul] |
| change (∫ x : UnitAddCircle × UnitAddCircle, B*(1-ψ x.2^2) |
| ∂AddCircle.haarAddCircle.prod AddCircle.haarAddCircle) = _ |
| rw [integral_fun_snd (μ := AddCircle.haarAddCircle (T := (1:ℝ))) |
| (ν := AddCircle.haarAddCircle) (fun a : UnitAddCircle => B*(1-ψ a^2))] |
| simp only [probReal_univ,one_smul,integral_const_mul] |
|
|
| lemma packet_neutral_cutoff_error {A : UnitAddCircle × UnitAddCircle → ℂ} |
| {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) {δ : ℝ} |
| (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (m : ℤ) : |
| (∫ x : TripleTorus, ‖packet m A x-(neutralCircleCutoff c x.2.2:ℂ)*packet m A x‖^2 |
| ∂tripleVolume) ≤ B*(4*δ) := |
| (packet_cutoff_error_uniform (continuous_neutralCircleCutoff c) |
| (neutralCircleCutoff_bounds c hδ) hB hAb m).trans |
| (mul_le_mul_of_nonneg_left (neutralCircleCutoff_error c hδ hδ') hB) |
|
|
| lemma uncut_retainedPassage_uniform_error |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {m : ℤ} |
| {H : ℕ → UnitAddCircle → UnitAddCircle} {χ : ℝ → UnitAddCircle → ℝ} |
| (hA : Continuous A) (hH : ∀ n, Continuous (H n)) (hχ : Continuous χ.uncurry) |
| {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) {δ : ℝ} |
| (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) (n : ℕ) (S : Finset ℤ) : |
| (∫ x : TripleTorus, |
| ‖packet m A x-retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) |
| (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) ≤ |
| 8*δ*B+2*(∫ x : TripleTorus, |
| ‖(neutralCircleCutoff c x.2.2:ℂ)*packet m A x- |
| retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) |
| (neutralCircleCutoff c) (passageChart (H n) x)‖^2 ∂tripleVolume) := by |
| let f := packet m A |
| let g := fun x : TripleTorus => (neutralCircleCutoff c x.2.2:ℂ)*f x |
| let h := fun x : TripleTorus => retainedPassage m (H n) A S (fun j => χ ((j:ℝ)/n)) |
| (neutralCircleCutoff c) (passageChart (H n) x) |
| have hf : Continuous f := continuous_packet hA |
| have hg : Continuous g := (Complex.continuous_ofReal.comp |
| ((continuous_neutralCircleCutoff c).comp continuous_snd.snd)).mul hf |
| have hh : Continuous h := |
| (continuous_retainedPassage (continuous_neutralCircleCutoff c) hχ hH hA n S).comp |
| (continuous_passageChart (hH n)) |
| have hi (v w : TripleTorus → ℂ) (hv : Continuous v) (hw : Continuous w) : |
| Integrable (fun x => ‖v x-w x‖^2) tripleVolume := |
| ((hv.sub hw).norm.pow 2).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have hfg := hi f g hf hg |
| have hgh := hi g h hg hh |
| calc |
| _ ≤ ∫ x, (2*‖f x-g x‖^2+2*‖g x-h x‖^2) ∂tripleVolume := by |
| apply integral_mono (hi f h hf hh) ((hfg.const_mul 2).add (hgh.const_mul 2)) |
| intro x |
| exact sq_norm_sub_le_two (f x) (g x) (h x) |
| _ = 2*(∫ x, ‖f x-g x‖^2 ∂tripleVolume)+2*(∫ x, ‖g x-h x‖^2 ∂tripleVolume) := by |
| rw [integral_add (hfg.const_mul 2) (hgh.const_mul 2),integral_const_mul,integral_const_mul] |
| _ ≤ _ := by |
| have hh := packet_neutral_cutoff_error hB hAb c hδ hδ' m |
| change (∫ x, ‖f x-g x‖^2 ∂tripleVolume) ≤ _ at hh |
| dsimp only [g,h] |
| nlinarith |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| section |
|
|
| /-! The actual neutral refresh: a single retained vector simultaneously |
| approximates the old physical packet and its old temporal spectrum. The |
| amplitude may depend on both slow coordinates. -/ |
| noncomputable section |
| open MeasureTheory Set Function Filter |
| open scoped Topology BigOperators |
| namespace ThreeTorus |
|
|
| theorem exact_neutral_passage |
| {A : UnitAddCircle × UnitAddCircle → ℂ} {A₀ : UnitAddCircle → ℝ → ℂ} |
| {Pr V : ℝ → ℝ} {m : ℤ} {Δ : ℕ → ℝ} {H : ℕ → UnitAddCircle → UnitAddCircle} |
| {o l u R : ℝ} |
| (hm : m ≠ 0) |
| (hH : ∀ n (x : ℝ), H n (x : UnitAddCircle) = ((n:ℝ)*(Pr x+Δ n*x) : UnitAddCircle)) |
| (hA : ∀ a x, x ∈ Ioc o (o+1) → A ((x : UnitAddCircle),a) = A₀ a x) |
| (hA₀ : ∀ a, ContDiff ℝ 2 (A₀ a)) (hAc : Continuous A₀.uncurry) |
| (hAd : Continuous (fun p : UnitAddCircle × ℝ => deriv (A₀ p.1) p.2)) |
| (hAr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun p : ℝ × ℝ => A₀ (p.2 : UnitAddCircle) p.1)) |
| (hsupp : ∀ a, support (A₀ a) ⊆ Ioc l u) |
| (hlo : o ≤ l) (huo : u ≤ o+1) (hlu : l ≤ u) (hR : 0 < R) |
| (hPr : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) Pr) |
| (hV : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) V) (hPV : deriv Pr = V) |
| (hdV : ∀ x ∈ Icc (l-2*R) (u+2*R), 0 < deriv V x) |
| (h0 : 0 ∉ (fun x => (m:ℝ)*V x) '' Icc l u) |
| (hHc : ∀ n, Continuous (H n)) (hAcont : Continuous A) (hΔt : Tendsto Δ atTop (𝓝 0)) |
| (hAs : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun p : ℝ × ℝ => A (p.1,p.2))) |
| {v : UnitAddCircle → ℝ} (hv : Measurable v) (hVe : ∀ x : ℝ, v (x:UnitAddCircle) = V x) |
| {B : ℝ} (hB : 0 ≤ B) (hAb : ∀ p, ‖A p‖^2 ≤ B) |
| {δ : ℝ} (c : LawCutoffs δ) (hδ : 0 < δ) (hδ' : δ < 1/8) |
| {χ : ℝ → UnitAddCircle → ℝ} |
| (hχs : CylinderSmooth χ.uncurry) |
| (hχc : Continuous χ.uncurry) (hcχ : HasCompactSupport χ.uncurry) (hzχ : ∀ a, χ 0 a = 0) |
| (hχ : ∀ s a, 0 ≤ χ s a ∧ χ s a ≤ 1) |
| (hχW : ∀ s a, stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s ≠ 0 → χ s a = 1) : |
| let pold := foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (l-R) (u+R)) |
| (fun x => (m:ℝ)*V x) (A₀ a) s ∂AddCircle.haarAddCircle) |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ L : ℝ, 0 < L → ∀ β : ℝ, |
| ∃ S : ℕ → Finset ℤ, |
| (∀ n, 0 < n → ∀ j ∈ S n, j ≠ 0) ∧ |
| (∀ n, 0 < n → ∀ j ∉ S n, ∀ a, χ ((j:ℝ)/n) a = 0) ∧ |
| ∀ η : ℝ, 0 < η → ∀ᶠ n : ℕ in atTop, |
| let ψ := neutralCircleCutoff c |
| let J := neutralJitter c L β |
| let φ := fun a => (((1+J a/n)/n:ℝ):UnitAddCircle) |
| let f := retainedPassage m (H n) A (S n) (fun j => χ ((j:ℝ)/n)) ψ |
| let p := exactRetainedPacketDensity (Ioo δ (1-δ)) (neutralJitterReal c L β) |
| ψ χ m H A (S n) n |
| (∫ x : TripleTorus, ‖packet m A x-f (passageChart (H n) x)‖^2 ∂tripleVolume) < |
| 8*δ*B+η ∧ |
| (∫ z, |p z-pold z| ∂AddCircle.haarAddCircle) < C/L+4*δ*B+η ∧ |
| ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) (fun x : ℝ => p x) ∧ |
| (∀ k : ℤ, (∫ x : TripleTorus, starRingEnd ℂ (f x)*f (twistIter φ k x) ∂tripleVolume) = |
| ∫ z, (p z:ℂ)*fourier k z ∂AddCircle.haarAddCircle) ∧ |
| (∀ j u a, ‖retainedAmplitude ψ χ m H A n j (u,a)‖ = |
| ‖retainedAmplitude ψ χ m H A n j (u,0)‖) := by |
| dsimp only |
| let ψ := neutralCircleCutoff c |
| let W := fun s a => stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) (A₀ a) s |
| let pold := foldCircleDensity (fun s => ∫ a, W s a ∂AddCircle.haarAddCircle) |
| let pcut := foldCircleDensity (fun s => ∫ a, stationaryWeight (Icc (l-R) (u+R)) |
| (fun x => (m:ℝ)*V x) (fun x => (ψ a:ℂ)*A₀ a x) s ∂AddCircle.haarAddCircle) |
| let F := stationaryCutoffLift (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) |
| (fun p : ℝ × ℝ => A₀ (p.2:UnitAddCircle) p.1) c.ψ |
| have hψc := continuous_neutralCircleCutoff c |
| have hψs : tsupport c.ψ ⊆ Ioc (0:ℝ) (0+1) ∩ Ioo δ (1-δ) := by |
| intro x hx |
| have hh := c.ψ_support hx |
| exact ⟨⟨by linarith [hh.1],by linarith [hh.2]⟩,hh⟩ |
| have heψ : ∀ x ∈ Ioc (0:ℝ) (0+1), c.ψ x = ψ (x:UnitAddCircle) := by |
| intro x hx |
| exact (lawPsi_eq c hδ ⟨hx.1.le,by simpa using hx.2⟩).symm |
| have hψsupport : ∀ x ∈ Ioc (0:ℝ) 1, ψ (x:UnitAddCircle) ≠ 0 → x ∈ Ioo δ (1-δ) := by |
| intro x hx hn |
| rw [← heψ x (by simpa using hx)] at hn |
| exact (hψs (subset_tsupport c.ψ hn)).2 |
| have hcut : (∫ z, |pcut z-pold z| ∂AddCircle.haarAddCircle) ≤ B*(4*δ) := by |
| have hs₁ : ∀ a, support (A₀ a) ⊆ Ioc o (o+1) := by |
| intro a x hx |
| have hh := hsupp a hx |
| exact ⟨lt_of_le_of_lt hlo hh.1,hh.2.trans huo⟩ |
| have hs₂ : ∀ a, support (A₀ a) ⊆ Ioo (l-R) (u+R) := by |
| intro a x hx |
| have hh := hsupp a hx |
| exact ⟨by linarith [hh.1],by linarith [hh.2]⟩ |
| have hb := oldStationaryMarginal_cutoff_l1 hAcont hAc hA hs₁ hs₂ hv hV |
| (fun x _ => hVe x) isOpen_Ioo Ioo_subset_Icc_self |
| (scaled_speed_good_branch hm hR hV hdV).1 (scaled_speed_good_branch hm hR hV hdV).2 |
| hψc (neutralCircleCutoff_bounds c hδ) hB hAb |
| exact hb.trans (mul_le_mul_of_nonneg_left (neutralCircleCutoff_error c hδ hδ') hB) |
| have hmarg : (fun s => ∫ a, F (a,s)) = fun s => ∫ a, |
| stationaryWeight (Icc (l-R) (u+R)) (fun x => (m:ℝ)*V x) |
| (fun x => (ψ a:ℂ)*A₀ a x) s ∂AddCircle.haarAddCircle := by |
| funext s |
| exact stationaryCutoffLift_marginal heψ |
| (fun x hx => (hψs (subset_tsupport c.ψ hx)).1) _ _ s |
| obtain ⟨C,hC,hsp⟩ := exact_neutral_circle_spectral_estimate hm hH hA hA₀ hAc hAd hAr |
| hsupp hlo huo hlu hR hPr hV hPV hdV h0 hHc hAcont hΔt isOpen_Ioo |
| c.ψ_smooth c.ψ_compact hψs heψ hψc hχc hcχ hzχ |
| (neutralCircleCutoff_bounds c hδ) hχ hχW (Q := 1) (by decide) 0 |
| refine ⟨C,hC,?_⟩ |
| intro L hL β |
| have hJs := neutralJitterReal_smooth c L β |
| have hJc := continuous_neutralJitter c L β |
| have heJ : ∀ a ∈ Ioo δ (1-δ), neutralJitterReal c L β a = L*a+β := |
| fun a ha => neutralJitterReal_affine c hδ L β ⟨ha.1.le,ha.2.le⟩ |
| have hJinj : InjOn (neutralJitterReal c L β) (Ioo δ (1-δ)) := by |
| intro a ha b hb hab |
| rw [heJ a ha,heJ b hb] at hab |
| nlinarith |
| have hJd : ∀ a ∈ Ioo δ (1-δ), deriv (neutralJitterReal c L β) a ≠ 0 := by |
| intro a ha |
| rw [neutralJitterReal_deriv c hδ L β ha] |
| exact hL.ne' |
| obtain ⟨S,hSN,hST,ht⟩ := hsp L hL β (neutralJitterReal c L β) (neutralJitter c L β) |
| heJ hJs hJc.measurable (fun x _ => neutralJitter_coe c L β x) |
| simp only [Nat.cast_one,mul_one,zero_add,inv_one,one_mul] at hSN hST ht |
| refine ⟨S,hSN,hST,?_⟩ |
| intro η hη |
| have hph := exact_physical_retainedPassage_tendsto hm hH hA hA₀ hAc hAd hsupp |
| hlo huo hlu hR hPr hV hPV hdV hHc hAcont hΔt hχc hχ |
| ψ hψc (neutralCircleCutoff_bounds c hδ) hχW S hST |
| have hpi (n : ℕ) (hn : 0 < n) := exactRetainedPacketDensity_integrable (m := m) hJs isOpen_Ioo hJinj hJd |
| hψc hχc hHc hAcont hn (hSN n hn) |
| have hpcuti : Integrable pcut AddCircle.haarAddCircle := by |
| have hreg := exact_cutoff_weight_regular hm hR hV hdV hAr |
| (fun a => (hsupp a).trans Ioc_subset_Icc_self) c.ψ_smooth c.ψ_compact h0 |
| have hi := compact_slice_integral_regular |
| ((hreg.1.comp (contDiff_snd.prodMk contDiff_fst)).continuous) |
| (hreg.2.1.comp_homeomorph (Homeomorph.prodComm ℝ ℝ)) |
| dsimp only [pcut] |
| rw [← hmarg] |
| exact integrable_foldCircleDensity hi.1.measurable hi.2 |
| have hpoldi : Integrable pold AddCircle.haarAddCircle := by |
| have hs₁ : ∀ a, support (A₀ a) ⊆ Ioc o (o+1) := fun a x hx => |
| ⟨lt_of_le_of_lt hlo (hsupp a hx).1,(hsupp a hx).2.trans huo⟩ |
| have hs₂ : ∀ a, support (A₀ a) ⊆ Ioo (l-R) (u+R) := by |
| intro a x hx |
| have hh := hsupp a hx |
| exact ⟨by linarith [hh.1],by linarith [hh.2]⟩ |
| exact (oldStationaryMarginal_map hAcont hAc hA hs₁ hs₂ hv hV |
| (fun x _ => hVe x) isOpen_Ioo Ioo_subset_Icc_self |
| (scaled_speed_good_branch hm hR hV hdV).1 (scaled_speed_good_branch hm hR hV hdV).2).1 |
| have hspec := ht (η/2) (half_pos hη) |
| change ∀ᶠ n : ℕ in atTop, (∫ z, |exactRetainedPacketDensity (Ioo δ (1-δ)) |
| (neutralJitterReal c L β) ψ χ m H A (S n) n z-foldCircleDensity (fun s => ∫ a, F (a,s)) z| |
| ∂AddCircle.haarAddCircle) < C/L+η/2 at hspec |
| rw [hmarg] at hspec |
| have hHl : ∀ n : ℕ, ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞) |
| (fun x => (n:ℝ)*(Pr x+Δ n*x)) := |
| fun _ => contDiff_const.mul (hPr.add (contDiff_const.mul contDiff_id)) |
| filter_upwards [hspec,hph.eventually (eventually_lt_nhds (half_pos hη)), |
| eventually_gt_atTop (0:ℕ)] with n hnsp hnph hn |
| refine ⟨?_,?_,?_,?_,?_⟩ |
| · have hh := uncut_retainedPassage_uniform_error (m := m) hAcont hHc hχc hB hAb c hδ hδ' n (S n) |
| exact hh.trans_lt (by linarith) |
| · exact (integral_abs_sub_triangle (hpi n hn) hpcuti hpoldi).trans_lt (by linarith) |
| · exact exactRetainedPacketDensity_smooth_lift hJs isOpen_Ioo hJinj hJd |
| c.ψ_smooth c.ψ_compact (fun x hx => (heψ x hx).symm) hψs hχs hHl hH hAs hn (hSN n hn) |
| · intro k |
| exact retainedPassage_spectral_moment hψc hχc hHc hAcont hψsupport hJc hJs |
| (fun x _ => neutralJitter_coe c L β x) isOpen_Ioo hJinj hJd hn (hSN n hn) k |
| · exact fun _ _ _ => retainedAmplitude_modulus _ _ _ _ _ _ _ _ _ |
|
|
| end ThreeTorus |
| end |
| end |
|
|
| end OAI |
|
|
